[Searches] Fix Binary Search bug with duplicate elements

[shuhao-alan-fan]

Describe your change:

This PR fixes a bug where the Binary Search algorithm returned incorrect results (or failed) when the input list contained duplicate elements.

I have updated the logic to handle duplicates correctly and added doctests to verify the fix.

Fixes #13886

• Add an algorithm?
• Fix a bug or typo in an existing algorithm?
• Add or change doctests? -- Note: Please avoid changing both code and tests in a single pull request.
• Documentation change?

Checklist:

• I have read CONTRIBUTING.md.
• This pull request is all my own work -- I have not plagiarized.
• I know that pull requests will not be merged if they fail the automated tests.
• This PR only changes one algorithm file. To ease review, please open separate PRs for separate algorithms.
• All new Python files are placed inside an existing directory.
• All filenames are in all lowercase characters with no spaces or dashes.
• All functions and variable names follow Python naming conventions.
• All function parameters and return values are annotated with Python type hints.
• All functions have doctests that pass the automated testing.
• All new algorithms include at least one URL that points to Wikipedia or another similar explanation.
• If this pull request resolves one or more open issues then the description above includes the issue number(s) with a closing keyword: "Fixes Binary Search gives wrong output for duplicate elements #13886".

[shuhao-alan-fan]

Hi maintainers, this PR fixes issue #13886 and all tests + pre-commit checks are passing.
Whenever you have time, I would appreciate a review.

Thank you!

[Copilot]

Pull request overview

This PR adds a new binary_search_with_duplicates function to handle binary search in sorted lists containing duplicate elements. The implementation uses lower_bound and upper_bound helper functions to find the range of all occurrences of the target value, addressing issue #13886.

Key changes:

• Adds binary_search_with_duplicates function that returns a list of all indices where the target occurs
• Implements lower_bound and upper_bound as nested helper functions using binary search
• Includes doctests covering single occurrences, multiple duplicates, and not-found cases

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[Copilot]

Pull request overview

Copilot reviewed 1 out of 1 changed files in this pull request and generated 6 comments.

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[Copilot]

Pull request overview

Copilot reviewed 1 out of 1 changed files in this pull request and generated 3 comments.

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