Regression in `1.0.13`: `session.send_and_wait` fails with `400` Bad Request in GitHub Actions (works on `1.0.11`)

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## Why are you seeing this issue?
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## Severity
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- [ ] sev1 - **Engineering must _respond_, when paged, within 1 business day**
- [x] sev2 - **Engineering must _respond_ to this issue within 2 business days**
- [ ] sev3 - **Engineering must _respond_ to this issue within 2 business days**

## Service
[copilot-sdk](https://catalog.githubapp.com/services?service=copilot-sdk)

## Context
_:warning: Kindly provide all the requested information to ensure a prompt and efficient response. :warning:_

**GitHub Support Plan**

- [ ] Top 20
- [x] Premium Plus
- [ ] Premium
- [ ] Enterprise
- [ ] Free/Pro/Team

**GitHub Product(s) Impacted (check appropriate boxes and provide GHES version if applicable)**

- [x] GitHub.com (including GHEC)
- [ ] GitHub Enterprise Server.
- [ ] Running version: x.y.z
- [ ] Recently upgraded from version: x.y.z (upgrade date: yyyy-mm-dd)
- [ ] Proxima (ghe.com)
- [ ] Customer tenant: -

**Customer Impact**

If applicable, list the impacted entities:
- [x] Enterprise: - Sabre Corporation
- [ ] Organization: -
- [ ] Repository: -
- [ ] User: -

**Zendesk Ticket(s), support bundles, and other relevant Issues and links**
- https://github.zendesk.com/agent/tickets/4735364

## Issue

> we have build some workflows/actions which use python/copilot-sdk to execute some code with copilot.
It was working fine until Friday (maybe longer, but we don't run it in weekends) but today we are getting errors.
We used GITHUB_TOKEN with copilot-requests: write persmission.

Using `1.0.11` works. On Friday we released `1.0.13` and that is confirmed that is causing the issue.

**To Reproduce**

Steps to reproduce the behavior:
1. Use the workflow and python from the ticket https://github.zendesk.com/agent/tickets/4735364
2. See the error
3. Force Copilot-sdk to be `1.0.11`
4. Now it works

**Expected behavior**

`1.0.13` not throwing `400` bad request.

**Screenshots**

If applicable, add screenshots to help explain your problem.

**Additional context**

[splunk](https://splunk.githubapp.com/en-US/app/gh_reference_app/search?earliest=1788760800&latest=1788778800&q=search%20index%3D*%20%224020%3A41139%3A1752DC9%3A1A47F55%3A6A9E7A01%22&display.page.search.mode=verbose&dispatch.sample_ratio=1&workload_pool=&sid=1788773903.89675_2DA2FCF5-3918-4AB7-BE2E-B8D81EDC01C6)

```
exception.message: request failed: checking server-to-server token: bad request: GitHub App Server-To-Server Tokens are not supported for this endpoint
```

That looks just weird. example request id in the splunk query. You can reproduce this easily.

cc:
@evanboyle
@patniko

**Support Squad(s)**: @github/support-squad-copilot @github/support-squad-c2c

Bắt đầu với Python workflow và các bước tái hiện trong Zendesk ticket 4735364, sau đó so sánh các phiên bản 1.0.11 và 1.0.13 của copilot-sdk liên quan đến session.send_and_wait. Sử dụng lỗi 400 và request ID được báo cáo để truy vết quá trình kiểm tra token server-to-server; được xem là hoàn tất khi GitHub Actions workflow chạy thành công trên 1.0.13 mà không có lỗi đó.

Do mô hình lập chỉ mục viết ra từ nội dung của issue.