def METHOD_NAME(self) -> None:
    """Registers parameterized routes for API Gateway user invocations."""
    if self.registered:
        LOG.debug("Skipped API Gateway route registration (routes already registered).")
        return
    self.registered = True
    LOG.debug("Registering parameterized API Gateway routes.")
    host_pattern = "<regex('[^-]+'):api_id><regex('(-vpce-[^.]+)?'):vpce_suffix>.execute-api.<regex('.*'):server>"
    self.router.add(
        "/",
        host=host_pattern,
        endpoint=self.invoke_rest_api,
        defaults={"path": "", "stage": None},
        strict_slashes=True,
    )
    self.router.add(
        "/<stage>/",
        host=host_pattern,
        endpoint=self.invoke_rest_api,
        defaults={"path": ""},
        strict_slashes=False,
    )
    self.router.add(
        "/<stage>/<path:path>",
        host=host_pattern,
        endpoint=self.invoke_rest_api,
        strict_slashes=True,
    )
    # add the localstack-specific _user_request_ routes
    self.router.add(
        "/restapis/<api_id>/<stage>/_user_request_",
        endpoint=self.invoke_rest_api,
        defaults={"path": ""},
    )
    self.router.add(
        "/restapis/<api_id>/<stage>/_user_request_/<path:path>",
        endpoint=self.invoke_rest_api,
        strict_slashes=True,
    )