aws_api_gateway_integration URI for kinesis

resource "aws_api_gateway_integration" "kinesis_integration" {
  rest_api_id          = aws_api_gateway_rest_api.shippeo_api_kinesis.id
  resource_id          = aws_api_gateway_resource.shippeo_api_resource_kinesis.id
  http_method          = aws_api_gateway_method.post_json_files_kinesis.http_method
  type                 = "AWS_PROXY"
  uri                  = aws_kinesis_stream.shippeo-kinesis-stream.arn
  integration_http_method     = "POST"
  depends_on = [
    aws_api_gateway_resource.shippeo_api_resource_kinesis
  ]
}

I want to create an API Gateway that sends data to a kinesis stream. I am trying the code above but I get this error:

Error creating API Gateway Integration: BadRequestException: AWS ARN for integration must contain path or action
│ 
│   with module.shippeo-api.aws_api_gateway_integration.kinesis_integration,
│   on ../../modules/shippeo/kinesis-api.tf line 62, in resource "aws_api_gateway_integration" "kinesis_integration":
│   62: resource "aws_api_gateway_integration" "kinesis_integration" {

I read in another SO answer that the format of the uri is supposed to be like this:

arn:aws:apigateway:{region}:{subdomain.service|service}:{path|action}/{service_api}

How can I change it to use it with kinesis? What is the action or service_api?

region = "eu-central-1"