ATATcH Alternating Treatment Plans for Advanced Cancer

Lung Cancer (Arms 1 and 2) * Patients must have histologically or cytologically confirmed stage IV NSCLC (includes M1a, M1b, and M1c stage disease, AJCC 8th edition). Patients with Stage IIIB and IIIC disease are eligible if they are not candidates for combined chemotherapy and radiation; such cases should be discussed in a multidisciplinary tumor board. * Eligible NSCLC tissue histologies will include squamous cell carcinoma (enrolled and treated in Arm 1), and nonsquamous histologies (e.g. adenocarcinoma, large cell carincoma, etc.; enrolled and treated in Arm 2). Patients with mixed squamous, e.g., adenosquamous, histology will be enrolled and treated on Study Arm 1. Patients with any evidence of Small Cell Carcinoma will be excluded from study participation. * Patients may have ANY PD-L1 expression Tumor Proportion Score (TPS) status. Tissue testing for PD-L1 is strongly recommended. If PD-L1 expression TPS is unevaluable or the testing could not be completed, the patient may still be eligible. * Patients must have measurable or non-measurable disease. The presence of malignant pleural fluid alone is sufficient to satisfy this eligibility criterion. Baseline imaging assessments and measurements used to evaluate all measurable or non-measurable sites of disease must be done within 4 weeks prior to study registration. NOTE: If patient receives pemetrexed, follow institutional guidelines to drain fluids. * Patients must be ≥ 18 years of age. * Patients must have an ECOG Performance Status of 0 to 2 * Patients must NOT have received the following: * Prior systemic chemotherapy or immunotherapy for advanced metastatic NSCLC. Patients treated with any prior checkpoint inhibitors for metastatic lung cancer are ineligible. Chemotherapy and immunotherapy for non-metastatic disease (e.g. adjuvant therapy) or immunotherapy for locally advanced Stage III disease is allowed if at least 6 months have elapsed between the last dose of the prior therapy and study registration. Local therapy, e.g. palliative radiation, is allowed as long as a period of 7 days has passed between completion of local therapy and study registration. Registration prior to treatment during the 7 days is allowed. Palliative radiation must be to non-target lesions. * Methotrexate (MTX) given in low doses for non-malignant conditions with last dose at least 14 days prior to date of registration will be allowed. Other low dose chemotherapeutics for non-malignant conditions will be considered, but review by the study chair is required. * For Arm 1 (Squamous Lung Cancer): Patients must not have pre-existing peripheral neuropathy that is ≥ Grade 2 by CTCAE v.5.0. Patients must not have known sensitivity to any component of carboplatin or paclitaxel or nabpaclitaxel. * Patients with known EGFR mutations (except exon 20 insertion), BRAF mutations (V600), MET Exon14 skipping or ALK or ROS1 translocations that can be treated with oral tyrosine kinase inhibitors are excluded. * Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression. Patients with asymptomatic new (at screening) or progressive brain metastases (active brain metastases at screening) or leptomeningeal disease are eligible if the treating physician determines that immediate CNS specific treatment is not required and is unlikely to be required during the first cycle of therapy. * Patients are eligible if off steroids for at least 7 days prior to protocol treatment. * Palliative radiation to non-target lesions (bone metastasis) is allowed if patient develops symptoms. * Anticonvulsants are allowed. * Patients with asymptomatic, sub-centimeter brain metastasis who at the discretion of investigators do not need immediate CNS directed therapies are eligible. * Patients with prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial. * Patients must not have known pre-existing and clinically active interstitial lung disease, or a known history of (non-infectious) pneumonitis that required steroids, or current pneumonitis. * Patients must not have significant gastrointestinal disorders with diarrhea as a major symptom (e.g. Crohn's disease, malabsorption, etc.) * Patients must not have history of auto-immune condition requiring ongoing or intermittent systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class II or better. * Patients must not have any other concomitant serious illness or organ system dysfunction that in the opinion of the investigator would either compromise patient safety or interfere with the evaluation of the safety of the study drug. * Patients must not receive any other investigational agents during the course of therapy. * Women must not be pregnant or breast-feeding due to potential harm to the fetus or infant from cytotoxic chemotherapy and the unknown risk of MK-3475 (pembrolizumab). Patients must also not expect to conceive or father children from the time of registration, while on study treatment, and until at least 120 days after the last dose of study treatment. All females of childbearing potential must have a blood test or urine study within 72 hours prior to registration to rule out pregnancy. A female of childbearing potential is any woman, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1) has achieved menarche at some point; 2) has not undergone a hysterectomy or bilateral oophorectomy; or 3) has not been naturally postmenopausal (amenorrhea following cancer therapy does not rule out childbearing potential) for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months). * Women of childbearing potential and sexually active males must use an accepted and effective method of contraception or abstain from sexual intercourse from time of registration, while on study treatment, and continue for 120 days after the last dose of study treatment. * Patient must have the ability to understand and the willingness to sign a written informed consent document. * Patients must meet the following laboratory values within 14 days of randomization: * ANC ≥ 1500/mm3 * Platelets ≥ 100,000/mm3 * Hgb \> 8 g/dL (Note: Patient may be transfused to meet this criteria) * PT/INR ≤ 1.5 * Or if patient on therapeutic anticoagulation with Warfarin, PT/INR ≤ 3.0 * Patients must have adequate liver function as determined by the following tests obtained within 14 days of randomization: * Total Bilirubin ≤ 1.5 mg/dL * SGOT (AST) \< 5X upper limit of normal (ULN) * SGPT (ALT) \< 5X upper limit of normal ULN) * Patients must have adequate renal function as determined by the following tests obtained within 14 days prior to randomization: Calculated creatinine clearance ≥ 45ml/min to be eligible to receive pemetrexed Serum creatinine ≤ 1.5X institutional upper limit of normal (ULN) * Patients must not have a known history of active tuberculosis (TB). * Patients must not have a diagnosis of immunodeficiency or receive systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of protocol treatment. * Patients must not have received a live vaccine within 30 days prior to randomization. Seasonal flu vaccines that do not contain live virus are permitted. COVID-19 vaccination per guidelines for cancer patients is permitted and encouraged. * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial. * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable or on suppressive therapy, if indicated. Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. Head and Neck Cancer (Arm 3) * Patient must have histologically confirmed recurrent/metastatic squamous cell carcinoma of the head and neck (HNSCC) (excluding SCC of salivary glands, nasopharynx and skin) that is considered incurable by local therapies. The eligible primary tumor locations include oropharynx, oral cavity, hypopharynx, and larynx. Unknown primary site will also be considered eligible. * Patients may have ANY PD-L1 expression Tumor Proportion Score (CPS) status. Tissue testing for PD-L1 IHC on samples demonstrating recurrent/metastatic disease is strongly recommended, though testing may be performed on initial diagnostic specimens. If PD-L1 expression CPS is unevaluable or the testing could not be completed, the patient will still be considered eligible. * Tumor expression of p16 by immunohistochemistry is highly desirable for patients with Oropharyngeal primaries. Positive p16 expression is defined as strong and diffuse nuclear and cytoplasmic staining in 70% or more of the tumor cells. If HPV status was previously tested using this method, no additional testing is required. If results are not available or are not possible patient will still be considered eligible. Note: Oral cavity, hypopharynx, and larynx cancer are not required to undergo HPV testing by p16 IHC as by convention these tumor locations are assumed to be HPV negative. * Patients must have measurable or non-measureable disease. The presence of malignant pleural fluid or bone disease alone is sufficient to satisfy this eligibility criterion. Baseline imaging assessments and measurements used to evaluate all measurable or non-measurable sites of disease must be done within 4 weeks prior to study registration. Tumor lesions situated in a previously irradiated area are considered measurable if progression has been demonstrated in such lesions. * Patient must be ≥ 18 years of age. * Patient must have an ECOG performance status 0-2. * Patients must NOT have received the following: * Prior systemic chemotherapy or immunotherapy for recurrent/metastatic head and neck squamous cell carcinoma. Patients treated with any prior checkpoint inhibitors for recurrent/metastatic head and neck cancer are ineligible. NOTE: Patients who have received prior chemotherapy or cetuximab with radiation for curative intent treatment of locally advanced head and neck cancer whose disease has progressed after at least 6 months will be eligible. * Methotrexate (MTX) given in low doses for non-malignant conditions with last dose at least 14 days prior to date of registration will be allowed. Other low dose chemotherapeutics for non-malignant conditions will be considered, but review by the study chair is required. * Patients who have endocrinopathies but are now stable on hormone supplementation and/or a daily prednisone dose of ≤ 10 mg (or equivalent doses of another glucocorticoid), will be permitted on this trial. * Patient must not have a severe infection within 4 weeks prior to randomization, including, but not limited to, hospitalization for complications of infection, bacteremia, or severe pneumonia. Patients must not have active tuberculosis. * Patient must not have a history of non-infectious pneumonitis requiring steroids at doses greater than or equal to 10 mg per day of prednisone or the equivalent on first line immunotherapy. * Patient must not have a history of solid organ transplantation or stem-cell transplant. * Patient must not be on immunosuppressive medication within 7 days prior to randomization except for: intranasal, inhaled, or topical steroids, local steroid injection, systemic corticosteroids at doses less than or equal to 10 mg per day of prednisone or the equivalent, or steroids used as premedication for hypersensitivity reactions. * Patient must not have an active autoimmune disease that requires systemic treatment within 2 years prior to randomization. Patients who are receiving replacement therapy for adrenal or pituitary insufficiency will not be excluded. * Patient must not have had a severe hypersensitivity reaction to any of the drug components used on this protocol or to chimeric or humanized antibodies or fusion proteins. * Patient must not have received any live vaccine within 30 days prior to randomization and while participating in the study. Live vaccines include, but are not limited to, the following: measles, mumps, rubella, chicken pox, yellow fever, rabies, BCG, and typhoid (oral) vaccine. Patients are permitted to receive inactivated vaccines and any non-live vaccines including those for the seasonal influenza and COVID-19 (Note: intranasal influenza vaccines, such as Flu-Mist® are live attenuated vaccines and are not allowed). If possible, it is recommended to separate study drug administration from vaccine administration by about a week (primarily, in order to minimize an overlap of adverse events). * Patient must not be pregnant or breast-feeding due to the potential harm to an unborn fetus and possible risk for adverse events in nursing infants with the treatment regimens being used. All patients of childbearing potential must have a blood test or urine study within 14 days prior to randomization to rule out pregnancy. A patient of childbearing potential is defined as anyone, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1) has achieved menarche at some point, 2) has not undergone a hysterectomy or bilateral oophorectomy; or 3) has not been naturally postmenopausal (amenorrhea following cancer therapy does not rule out childbearing potential) for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months). * Patients must not expect to conceive or father children by using accepted and effective method(s) of contraception or by abstaining from sexual intercourse for the duration of their participation in the study and for 120 days after last dose of study therapy. * Patient must have the ability to understand and the willingness to sign a written informed consent document. * Patient must have adequate organ and marrow function as defined below (these labs must be obtained ≤ 14 days prior to protocol randomization): * Absolute neutrophil count (ANC) ≥ 1,500/mcL * Platelets ≥ 100,000/mcL * Hgb \> 8 g/dL (Note: Patient may be transfused to meet this criteria) * Total bilirubin ≤1.5xULN OR Direct bilirubin ≤ULN for subjects with total bilirubin levels \>1.5xULN * AST (SGOT)/ALT (SGPT) ≤ 2.5 × institutional ULN (\< 5.0 x institutional ULN if hepatic metastases present) * Creatinine ≤ 1.5 x institutional ULN OR Measured or Calculated Creatinine Clearance ≥60 mL/min for subject with creatinine levels \>1.5x institutional ULN * PT/INR ≤ 1.5 * Or if patient on therapeutic anticoagulation with Warfarin, PT/INR ≤ 3.0 * Patients with uncontrolled or symptomatic hypercalcemia (ionized calcium \> 1.5 mmol/L, calcium \> 12 mg/dL) must have their calcium levels corrected prior to randomization. * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months of randomization are eligible for this trial. * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. * Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression. Patients must not have untreated brain metastases or leptomeningeal disease. * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial. * Patient must not have significant cardiovascular disease (such as New York Heart Association Class II or greater cardiac disease, myocardial infarction, or cerebrovascular accident) within 3 months prior to randomization, or unstable arrhythmia or unstable angina at the time of randomization. * Patient must not receive any other chemotherapy, immunotherapy, antitumor hormonal therapy (excluding contraceptives and replacement steroids), radiation therapy, or experimental medications while on protocol treatment. Symptomatic lesions (e.g., bone metastases or metastases causing nerve impingement) amenable to palliative radiotherapy should be treated prior to randomization and patients must be recovered from the effects of radiation (there is no required minimum recovery period). * Patient must not have any other disease, metabolic dysfunction, physical examination finding, or clinical laboratory finding that contraindicates the use of the agents used in this protocol, may affect the interpretation of the results, or may render the patient at high risk from treatment complications.