Use the default token when updating dependencies

[a-maurice]

Description

Provide details of the change, and generalize the change in the PR title above.

Running into permission problems with the token, so changing the logic to use the default token when making the PR. Every other step still uses the previous token though, since that works for the other steps.

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Testing

Describe how you've tested these changes. Link any manually triggered Integration tests or CPP binary SDK Packaging Github Action workflows, if applicable.

https://github.com/firebase/firebase-cpp-sdk/actions/runs/13643145191

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Type of Change

Place an x the applicable box:

• Bug fix. Add the issue # below if applicable.
• New feature. A non-breaking change which adds functionality.
• Other, such as a build process or documentation change.

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Notes

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• Read the contribution guidelines CONTRIBUTING.md.
• Changes to the public API require an internal API review. If you'd like to help us make Firebase APIs better, please propose your change in a feature request so that we can discuss it together.