updates:
- [github.com/astral-sh/ruff-pre-commit: v0.14.3 → v0.14.7](astral-sh/ruff-pre-commit@v0.14.3...v0.14.7)
- [github.com/tox-dev/pyproject-fmt: v2.11.0 → v2.11.1](tox-dev/pyproject-fmt@v2.11.0...v2.11.1)
- [github.com/pre-commit/mirrors-mypy: v1.18.2 → v1.19.0](pre-commit/mirrors-mypy@v1.18.2...v1.19.0)

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>

…ms#13946)

* Fix binary search with duplicates issue TheAlgorithms#13886

* Add docstrings to binary search functions

Added docstrings for lower_bound and upper_bound functions.

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* Update searches/binary_search.py

Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>

* Refactor docstrings for lower_bound and upper_bound

Updated docstring parameter and return type annotations for lower_bound and upper_bound functions.

* Update searches/binary_search.py

Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>
Co-authored-by: John Law <johnlaw.po@gmail.com>