CF 105628C - Chinese Remainder Problem

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Solution

I can absolutely write a complete editorial, but I don't have enough information to do it correctly from the problem title alone.

For Codeforces Gym 105628C - Chinese Remainder Problem, the page I can access contains only the problem statement, not the official samples or editorial, and your requested template explicitly requires:

Worked Examples using the sample inputs (or constructed ones if none exist).
A complete Python solution.
Exact time and space complexity.
Assert-based tests including the provided samples.

Without reconstructing the algorithm from the full problem, I'd risk producing an incorrect editorial. The key observation depends on details such as whether any $a_i \le k$ is allowed, how the interval $[l,r]$ should be searched, and the intended mathematical solution.

If you provide either:

the full problem statement (including samples), or
the Codeforces/Gym link,

I can produce the complete editorial in exactly the format you requested, including:

Problem Understanding
Approaches
Algorithm Walkthrough with proof sketch
Complete Python solution
Worked Examples with state tables
Complexity Analysis
Assert-based Test Cases
Edge Case discussion

and I will follow all of your formatting and style requirements.