refactor: Rework client timeout #191

Job	Run time
Lint check / Lint check (3.9)	5s
Type check / Type check (3.9)	13s
Unit tests / Unit tests (ubuntu-latest, 3.9)	24s
Type check / Type check (3.10)	12s
Lint check / Lint check (3.10)	7s
Type check / Type check (3.11)	12s
Unit tests / Unit tests (ubuntu-latest, 3.10)	19s
Lint check / Lint check (3.11)	6s
Unit tests / Unit tests (ubuntu-latest, 3.11)	19s
Type check / Type check (3.12)	14s
Lint check / Lint check (3.12)	7s
Unit tests / Unit tests (ubuntu-latest, 3.12)	24s
Type check / Type check (3.13)	13s
Lint check / Lint check (3.13)	14s
Unit tests / Unit tests (ubuntu-latest, 3.13)	24s
Unit tests / Unit tests (windows-latest, 3.9)	44s
Unit tests / Unit tests (windows-latest, 3.10)	41s
Unit tests / Unit tests (windows-latest, 3.11)	51s
Unit tests / Unit tests (windows-latest, 3.12)	53s
Unit tests / Unit tests (windows-latest, 3.13)	50s
Async dostrings check / Async docstrings check	14s
Docs check / Docs check	1m 26s
Integration tests / Integration tests (3.9)	12s
Integration tests / Approve integration tests for PRs from forks	0s
Integration tests / Integration tests (3.13)	16s
Integration tests / Integration tests for PRs from forks	0s
	9m 40s

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