Convert MakeFile docker run to Cloud Build - docker

Question

I have a repo that contains a Dockerfile and a MakeFile, usually what I do is run make && make mkdocs-serve in the local repo folder and the MakeFile takes care of 1) Building the image, 2) Building MKDocs (static site builder) and 3) Run MKDocs server at 127.0.0.1:8789
Here is what the MakeFile looks like:
#vars
IMAGENAME=my_mkdocs
USER=$(id -u)
.DEFAULT_GOAL := all
docker-clean:
-#docker rm ${IMAGENAME}
docker-build:
#docker build -t ${IMAGENAME}:v1 --build-arg=USER=${USER} .
mkdocs-build:
#docker run -it -v $(CURDIR):/home/mkdocs ${IMAGENAME}:v1 new /build
# #docker run -it --name ${IMAGENAME} -p 8789:80 ${IMAGENAME}:v1
mkdocs-serve:
#docker run --rm -it -v $(CURDIR):/home/mkdocs -p 8789:8000 ${IMAGENAME}:v1 mkdocs serve --dev-addr 0.0.0.0:8000
all: docker-clean docker-build
Now I want to build and run the Docker image using Cloud Build and Cloud Run, and I'm having difficulty converting MakeFile to Cloud Build syntax:
mkdoc-build:
I think the original bash syntax is:
docker run -it -v /home/mkdocs my_mkdocs:v1 new /build
Here is the Cloud Build syntax I think might work:
# Build MKDocs
- name: 'bash'
args: ['docker', 'run', '-it', '-v', '/home/mkdocs', 'my_mkdocs:v1', 'new', '/build']
mkdocs-serve:
I think the original bash syntax is:
docker run --rm -it -v /home/mkdocs -p 8789:8000 my_mkdocs:v1 mkdocs serve --dev-addr 0.0.0.0:8000
Here is the Cloud Build syntax I think might work:
# Build MKDocs
- name: 'bash'
args: ['docker', 'run', '--rm', '-it', '-v', '/home/mkdocs', '-p', '8789:8000', 'my_mkdocs:v1', 'mkdocs', 'serve', '--dev-addr', '0.0.0.0:8000']
Does this make sense to you? I'm sure this works on my local machine, but I feel they are wrong as there is no deployment to Cloud Run. There is indeed some documentation of how to call Cloud Run from Cloud Build:
- name: 'gcr.io/google.com/cloudsdktool/cloud-sdk'
entrypoint: gcloud
args: ['run', 'deploy', 'SERVICE-NAME', '--image', 'gcr.io/PROJECT_ID/IMAGE', '--region', 'REGION']
images:
- gcr.io/PROJECT_ID/IMAGE
But this seems to be a bit far from what the bash commands are.