Adaptive Tyrosine Kinase Inhibitor (TKI) Therapy In Patients With Thyroid Cancer

Inclusion Criteria: * All histologically or cytologically confirmed diagnosis of thyroid cancer, other than anaplastic or stromal-cell derived cancers. * Participants with differentiated thyroid cancers (DTC) must have negative thyroglobulin antibodies. * Measurable disease meeting the following criteria and confirmed by central radiographic review: * At least 1 lesion of ≥ 1.0 centimeter (cm) in the longest diameter for a non- lymph node or ≥ 1.5 cm in the short-axis diameter for a lymph node which is serially measurable according to Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 using computerized tomography/magnetic resonance imaging (CT/MRI). If there is only one target lesion and it is a non-lymph node, it should have a longest diameter of ≥ 1.5 cm. * Lesions that have had external beam radiotherapy (EBRT) or locoregional therapies such as radiofrequency (RF) ablation must show evidence of progressive disease (substantial size increase of ≥ 20%) within 12 months to be deemed a target lesion. * Participants must show evidence of disease progression comparing (a) scan in screening and (b) historical scan obtained within 12 months prior to signing informed consent, according to RECIST 1.1 assessed and confirmed by central radiographic review of CT and/or MRI scans. * Participants with DTC must not be eligible for possible curative surgery and must be radioiodine (RAI)-refractory / resistant as defined by at least one of the following: * One or more measurable lesions that do not demonstrate iodine uptake on any radioiodine scan * One or more measurable lesions that has progressed by RECIST 1.1 within 12 months of RAI therapy, despite demonstration of radioiodine avidity at the time of that treatment by pre- or post-treatment scanning. * Disease progression in a patient that has received a cumulative activity of RAI of ≥ 550 millicuries (mCi) (22 gigabecquerels), with the last RAI dose administered at least 6 months prior to study entry. * Otherwise deemed not a candidate for further RAI therapy by a multidisciplinary tumor board within 60 days of enrollment. * Participants with DTC must be receiving thyroxine suppression therapy and thyroid stimulating hormone (TSH) should not be elevated (TSH should be ≤ 0.1 mU/L). * "Measurable" tumor marker (non-stimulated thyroglobulin \>10 ng/mL or CEA\>10 ng/ML in patients with DTC; or serum basal calcitonin \>10 pg/mL in patients with MTC) * Participants may have received prior multi-kinase targeted therapy except the TKI used in this trial. For example, patients getting Lenvatinib on this study may have been previously treated with Sorafenib, Vandetanib, Sunitinib, Pazopanib, etc. Each of the TKI targeted agents will be counted individually, regardless of the duration of its administration. * Participants with known brain metastases who have completed whole brain radiotherapy, stereotactic radiosurgery or complete surgical resection, will be eligible if they have remained clinically stable, asymptomatic for 30 days. * All chemotherapy or radiation related toxicities must have resolved to \< Grade 2 severity per Common Terminology Criteria for Adverse Events (CTCAE v 5.0), except alopecia infertility, anemia (see separate criteria) and any toxicities deemed irreversible by the treating physician. * Participants must have an Eastern Cooperative Oncology Group (ECOG) Performance Status of 0 - 2. * Adequately controlled blood pressure (BP) with or without antihypertensive medications, defined as BP ≤ 150/90 mm Hg at screening and no change in antihypertensive medications within 1 week prior to Cycle 1/Day 1 * Adequate renal function defined as calculated creatinine clearance ≥ 30 mL/min (using Cockcroft/Gault formula) * Adequate bone marrow function * Adequate liver function * Males or females age ≥ 18 years at the time of informed consent * Females must not be breastfeeding or pregnant at Screening or Baseline. * Males and females must follow all contraception guidelines as outlined in the protocol guidelines. * Voluntary agreement to provide written informed consent and the willingness and ability to comply with all aspects of the protocol Exclusion Criteria: * Participants who have received any anticancer treatment (including Chinese herbal medicine specified for the treatment of tumor) within 21 days or any investigational agent within 30 days prior to the first dose of study drug. This does not apply to the use of TSH-suppressive thyroid hormone therapy. * Major surgery within 21 days prior to the first dose of study drug. * Palliative radiation therapy within 14 days prior to the first dose of study drug. * Participants having \> 30 mg/mL urine protein on urine dipstick testing (Participants with urine protein \< 1 g/24 hour (h) will be eligible). * Gastrointestinal malabsorption or any other condition that in the opinion of the investigator might affect the absorption of Lenvatinib, Sorafenib, Cabozantinib, or Vandetanib. * Significant cardiovascular impairment: history of (a) congestive heart failure greater than New York Heart association (NYHA) Class II, (b) unstable angina, (c) myocardial infarction, (d) stroke, or (e) cardiac arrhythmia associated with impairment within 6 months of the first dose of study drug. * Bleeding or thrombotic disorders (Treatment with low molecular weight heparin is allowed). * Radiographic evidence of major blood vessel invasion/infiltration. * Active hemoptysis (bright red blood of at least 0.5 teaspoon) within 21 days prior to the first dose of study drug. * Active infection (any infection requiring systemic treatment). * Active malignancy (except for DTC/MTC or definitively treated basal or squamous cell carcinoma of the skin, or carcinoma in-situ of the cervix or bladder) within the past 24 months. * Known intolerance to any of the study drugs (or any of the excipients). * Any medical or other condition which, in the opinion of the investigator, would preclude participation in a clinical trial. * Females who are pregnant or breastfeeding. * Participants who are taking prohibited medications outlined in protocol documentation.