[BUG] claude-code has incorrect $PATH

Bug Description

Running the claude-code action in a Github workflow that uses a custom image (not ubuntu:latest) that has a custom $PATH value, this value is not reflected by claude.

Steps to Reproduce

1. create a Github workflow job that uses a custom container image with custom tools pre-installed and available on its $PATH.

jobs:
  claude:
    runs-on: [self-hosted, foo]
    container:
      image: golang:1.24-bullseye
...

1. ensure allowed tools include allowed_tools: "Bash,Read,Edit,Write,Grep,Task,Replace"
2. ask claude to: "echo the $PATH and echo the go version"

Expected Behavior

• Running echo $PATH should show the complete path available to claude, e.g. /go/bin:/usr/local/go/bin:/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin
• Running go version should output the go version installed in the container, e.g. go version go1.24.3 linux/amd64

Actual Behavior

• Running echo $PATH outputs: /usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin
• Running go version outputs: go: command not found