Editing Openai/6943ca25-5f58-8006-882b-41249eb1ca7a (section)

==== I derived and verified a linear-time DP in NNN for a single mmm, but that’s too slow for many distinct mmm. ====
However, because cold positions jump by ≈m\approx m≈m, the number of cold values ≤10610^6106 is only about 106/(m+1)10^6/(m+1)106/(m+1). Summed over many different mmm, this behaves like a harmonic series and is actually feasible if we build only cold positions, not full DP.

===== For each distinct mmm appearing in the input: =====
# If n≤mn\le mn≤m: immediate win → YES.
# Else if v2(m+1)v_2(m+1)v2(m+1) even: use modulo rule.
# Else (odd case): generate cold positions starting at m+1m+1m+1 by testing whether c+(m+1)c+(m+1)c+(m+1) is cold (else c+(m+2)c+(m+2)c+(m+2)) until exceeding the maximum queried nnn for that mmm. Store cold positions in a boolean array (or hash set). Then answer each query by lookup.

Because consecutive cold positions differ by m+1m+1m+1 or m+2m+2m+2, you only generate O(#cold) per mmm, which is about 106/(m+1)10^6/(m+1)106/(m+1). Across up to 10410^4104 queries, this is typically fine in 4 seconds in C++.