def METHOD_NAME(platform: PlatformName) -> set[Architecture]:
    native_machine = platform_module.machine()
    # Cross-platform support. Used for --print-build-identifiers or docker builds.
    host_platform: PlatformName = (
        "windows"
        if sys.platform.startswith("win")
        else ("macos" if sys.platform.startswith("darwin") else "linux")
    )
    native_architecture = Architecture(native_machine)
    # we might need to rename the native arch to the machine we're running
    # on, as the same arch can have different names on different platforms
    if host_platform != platform:
        for arch_synonym in ARCH_SYNONYMS:
            if native_machine == arch_synonym.get(host_platform):
                synonym = arch_synonym[platform]
                if synonym is None:
                    # can't build anything on this platform
                    return set()
                native_architecture = Architecture(synonym)
    result = {native_architecture}
    if platform == "linux" and Architecture.x86_64 in result:
        # x86_64 machines can run i686 containers
        result.add(Architecture.i686)
    if platform == "windows" and Architecture.AMD64 in result:
        result.add(Architecture.x86)
    return result