docs: add prompt client/server example to address #498 #1901

Job	Run time
checks / readme-snippets	13s
checks / pre-commit	47s
checks / test (3.11, lowest-direct, ubuntu-latest)	59s
checks / test (3.12, lowest-direct, windows-latest)	3m 39s
checks / test (3.11, locked, windows-latest)	2m 42s
checks / test (3.11, lowest-direct, windows-latest)	2m 52s
checks / test (3.10, lowest-direct, ubuntu-latest)	1m 0s
checks / test (3.12, lowest-direct, ubuntu-latest)	1m 2s
checks / test (3.10, locked, ubuntu-latest)	50s
checks / test (3.12, locked, ubuntu-latest)	58s
checks / test (3.13, lowest-direct, ubuntu-latest)	53s
checks / test (3.11, locked, ubuntu-latest)	53s
checks / test (3.10, lowest-direct, windows-latest)	2m 47s
checks / test (3.10, locked, windows-latest)	2m 31s
checks / test (3.12, locked, windows-latest)	3m 28s
checks / test (3.14, lowest-direct, windows-latest)	2m 56s
checks / test (3.14, locked, windows-latest)	3m 2s
checks / test (3.13, lowest-direct, windows-latest)	2m 31s
checks / test (3.13, locked, ubuntu-latest)	49s
checks / test (3.13, locked, windows-latest)	2m 38s
checks / test (3.14, locked, ubuntu-latest)	2m 55s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 21s
all-green	4s
	41m 50s

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