Study of MK-1697 in Participants With Advanced Solid Tumors (MK-1697-001)

Inclusion Criteria: * For Part A; has a histologically- or cytologically-confirmed advanced/metastatic solid tumor and has received, been intolerant to, or been ineligible for all treatments known to confer clinical benefit * For Part B: has 1 of the following histologically or cytologically confirmed tumor types that are anti-programmed cell death protein 1 (anti PD-1)/anti-programmed death-ligand 1 (anti PD-L1) treatment naive: * CRC originating in either the colon or rectum that is locally advanced unresectable or metastatic (ie, Stage IV) and that has received, and progressed on, all available standard-of-care therapies including fluoropyrimidine, oxaliplatin, and irinotecan * HNSCC that is considered incurable by local therapies. The eligible primary tumor locations are oropharynx, oral cavity, hypopharynx, and larynx. Participants may not have a primary tumor site of nasopharynx (any histology). Also, participants must have progressed after receiving platinum-containing systemic therapy * Has measurable disease by Response Evaluation Criteria in Solid Tumors, version 1.1 (RECIST 1.1) * Has an evaluable baseline tumor sample (either a recent or archival) for analysis * Has an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Has central venous access (eg, portacath, Hickman line, or peripherally inserted central catheter \[PICC\] line) currently inserted or be considered medically fit for and willing to undergo the insertion of such a device * Is not pregnant or breastfeeding * Female participants of childbearing potential must agree to use contraception during the treatment period and for at least 120 days after the last dose of study treatment * Male participants must agree to use contraception during the treatment period and for at least 120 days after the last dose of study treatment and refrain from donating sperm during this period Exclusion Criteria: * Has a history of a second malignancy, unless potentially curative treatment has been completed with no evidence of malignancy for 3 years with the exception of participants who underwent successful definitive resection of basal cell carcinoma of the skin, superficial bladder cancer or in situ cervical cancer, or other in-situ cancers * Has clinically active central nervous system (CNS) metastases and/or carcinomatous meningitis * Has had a severe hypersensitivity reaction to treatment with any monoclonal antibody and/or components of the study treatment * Has an active infection requiring therapy * Has a history of interstitial lung disease * Has a history of (noninfectious) pneumonitis that required steroids or current pneumonitis * Has an active autoimmune disease that has required systemic treatment in the past 2 years * Has known human immunodeficiency virus (HIV) and/or Hepatitis B or C infections, or known to be positive for Hepatitis B antigen/Hepatitis B virus deoxyribonucleic acid (DNA) or Hepatitis C Antibody or ribonucleic acid (RNA) * Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the study, interfere with participation, make administration of the study treatments hazardous, or make it difficult to monitor adverse effects in the opinion of the treating investigator * Has a history or current evidence of severe cardiovascular disease, ie, arrhythmias requiring chronic treatment, congestive heart failure (New York Heart Association \[NYHA\] Class III or IV) or symptomatic ischemic heart disease. * Has a known psychiatric or substance abuse disorder that would interfere with the participant's ability to cooperate with the requirements of the trial * Is pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the study, starting with the screening visit through 120 days after the last dose of study treatment * Has not fully recovered from any effects of major surgery without significant detectable infection. Surgeries that required general anesthesia must be completed at least 2 weeks before first study treatment administration. Surgery requiring regional/epidural anesthesia must be completed at least 72 hours before first study treatment administration and participants should be recovered * Has known microsatellite instability (MSI) high or mismatch repair genes (MMR) deficient colorectal cancer. If a participant's MSI status is unknown, a paired blood sample for MSI in addition to biomarker testing is required to determine MSI status retrospectively (for the CRC expansion cohort only) * Has a positive pregnancy test within 72 hours before the first dose of study treatment * Has had chemotherapy, definitive radiation, or biological cancer therapy within 4 weeks (2 weeks for palliative radiation) prior to the first dose of study therapy, or has not recovered to Common Terminology Criteria for Adverse Events (CTCAE) Grade 1 or better from any adverse events that were due to cancer therapeutics administered more than 4 weeks earlier * Has received prior therapy with an anti-Lymphocyte-activation gene 3 (LAG-3) agent * Has received a live vaccine within 30 days prior to the first dose of study drug * Has undergone a prior stem cell or bone marrow transplant within the last 5 years * Is expected to require any other form of antineoplastic therapy while on study * Is currently participating in or has participated in a study of an investigational agent or has used an investigational device within 4 weeks prior to the first dose of study treatment