Master Protocol to Assess Safety and Dose of First Time in Human Next Generation Engineered T Cells in NY-ESO-1 and/or LAGE-1a Positive Advanced Solid Tumors

Eligibility Criteria: Inclusion criteria: * Participant must be \>=18 years of age and weighs ≥40 kg on the day of signing informed consent * Participant must be positive for HLA-A\*02:01, HLA-A\*02:05, and/or HLA-A\*02:06 alleles * Participant's tumor must have tested positive for NY-ESO-1 and/or LAGE-1a expression by a GSK designated laboratory * Performance status: Eastern Cooperative Oncology Group of 0-1 * Participant must have adequate organ function and blood cell counts 7 days prior to leukapheresis * Participant must have measurable disease according to RECIST v1.1. Additional criteria for participants with SS/ MRCLS: * Participant has advanced (metastatic or unresectable) SS or MRCLS confirmed by local histopathology with evidence of disease-specific translocation * Participant has completed at least one standard of care (SOC) treatment including anthracycline containing regimen unless intolerant to or ineligible to receive the therapy. Participants who are not candidates to receive anthracycline should have received ifosfamide unless also intolerant to or ineligible to receive ifosfamide. Participants who received neoadjuvant/adjuvant anthracycline or ifosfamide based therapy and progressed will be eligible Additional criteria for participants with non-small cell lung cancer (NSCLC): * Participant has Stage IV NSCLC as confirmed by histology or cytology * Prior therapies for participants lacking actionable genetic aberrations (i.e., wild type), per National Comprehensive Cancer Network (NCCN) guidelines: participant has been previously treated with or is intolerant to programmed death receptor-1 (PD)-1/Programmed death ligand 1 (PD-L1) checkpoint blockade therapy and has been previously treated with or is intolerant to a platinum-based chemotherapy. Adjuvant therapy will count as a regimen if completed within 6 months before relapse. Or for participants that harbors an actionable genetic aberration (e.g. BRAF, anaplastic lymphoma kinase \[ALK\]/ c-ros oncogene 1 \[ROS1\] etc.), per NCCN guidelines: participants has been previously treated with or is intolerant to SOC therapy, including targeted therapy, as recommended by NCCN or equivalent country-level guidelines (European Society for Medical Oncology \[ESMO\], National Institute for Health \& Care Excellence \[NICE\]) . Or Investigator has decided that additional lines of SOC therapy after the first line are not in the participant's best interest. Exclusion criteria: * Central nervous system (CNS) metastases, with certain exceptions for CNS metastases in NSCLC as specified in the protocol * Any other prior malignancy that is not in complete remission * Clinically significant systemic illness * Prior or active demyelinating disease * History of chronic or recurrent (within the last year prior to leukapheresis) severe autoimmune or immune mediated disease requiring steroids or other immunosuppressive treatments * Previous treatment with genetically engineered NY-ESO-1-specific T cells, NY-ESO-1 vaccine or NY-ESO-1 targeting antibody * Prior gene therapy using an integrating vector * Previous allogeneic hematopoietic stem cell transplant within the last 5 years or solid organ transplant * Washout periods for prior radiotherapy and systemic chemotherapy must be followed * Major surgery within 4 weeks prior to lymphodepletion * Pregnant or breastfeeding females