claude-code-action 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
...
```
2. ensure allowed tools include `allowed_tools: "Bash,Read,Edit,Write,Grep,Task,Replace"`
3. ask claude to: "echo the $PATH and echo the go version" (via a GH issue)

## 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`