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
_Note, for Sev1 issues always page the team first, using the PagerDuty information in the [service catalog](https://catalog.githubapp.com/services) and the command `.pager trigger N/A `. Opening an issue for a Sev2 incident can come later (and can be done by the available Escalation Engineer)._

- [ ] 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

先从 Zendesk 工单 4735364 中的 Python workflow 和复现步骤开始，然后比较 copilot-sdk 版本 1.0.11 和 1.0.13 在 session.send_and_wait 附近的差异。使用报告的 400 错误和请求 ID 追踪服务器到服务器的令牌检查；当 GitHub Actions workflow 在 1.0.13 上成功运行且不再出现该错误时，即视为完成。

由索引模型根据 Issue 内容生成。