NODE_ENV environment variable in Github action is not changed in separated jobs

I've separated Github action jobs in two.

One is for setting GITHUB_ENV variables, named set-env-variables and another is for deployment, named deploy.

However every variable except "NODE_ENV" is set perfectly in deploy job as I intended, and "NODE_ENV" is remained to be a default value.

Here is my simplified workflow file.

name: Deploy

on:
  push:
    branches:
      - main
      - staging

defaults:
  run:
    shell: bash

env:
  NODE_ENV: development
  DEPLOY_ENV: development

jobs:
  set-env-variables:
    runs-on: ubuntu-latest
    steps:
      - name: 'Check Pushed Branch'
      run: |
        BRANCH=$(echo ${GITHUB_REF##*/})

        if [ ${BRANCH} = staging ]; then
          echo "DEPLOY_ENV=staging" >> $GITHUB_ENV
          echo "NODE_ENV=staging" >> $GITHUB_ENV
        fi        

  deploy:
    runs-on: ubuntu-latest
    steps:
      - name: 'Deploy by environment'
      run: |
      echo $NODE_ENV
      echo $DEPLOY_ENV
      # do some deploying jobs...      

when pushing to staging branch, Deploy by environment step in deploy job shows the result below.

development
staging

I have no idea why NODE_ENV hasn't changed, but every other variables have changed well.

Mind the gap:

> You can make an environment variable available to any subsequent steps in a workflow job by defining or updating the environment variable and writing this to the GITHUB_ENV environment file.

(from : Setting an environment variable)

You have a different job.

Use GITHUB_OUTPUT instead:

> Sets a step's output parameter. Note that the step will need an id to be defined to later retrieve the output value.

(from: Setting an output parameter)

• Example of setting an output parameter

      - name: Set color
        id: random-color-generator
        run: echo "SELECTED_COLOR=green" >> "$GITHUB_OUTPUT"
      - name: Get color
  
        run: echo "The selected color is ${{ steps.random-color-generator.outputs.SELECTED_COLOR }}"
  

• Defining outputs for jobs - Create a map of outputs for your jobs.

• Example: Defining outputs for a job

  jobs:
    job1:
      runs-on: ubuntu-latest
      # Map a step output to a job output
      outputs:
        output1: ${{ steps.step1.outputs.test }}
        output2: ${{ steps.step2.outputs.test }}
      steps:
        - id: step1
          run: echo "test=hello" >> "$GITHUB_OUTPUT"
        - id: step2
          run: echo "test=world" >> "$GITHUB_OUTPUT"
    job2:
      runs-on: ubuntu-latest
      needs: job1
      steps:
        - env:
            OUTPUT1: ${{needs.job1.outputs.output1}}
            OUTPUT2: ${{needs.job1.outputs.output2}}
          run: echo "$OUTPUT1 $OUTPUT2"
  

• needs context

• Example usage of the needs context

Cf. GitHub Action - Define Workflow Level Environment Variable Between Jobs