Maximum Total Damage With Spell Casting

Aggregate before optimizing

For the usual spell-casting version, using a power prevents choosing powers within a stated numeric distance. First count equal values and turn each unique power p into reward p * frequency[p]. Sort the unique powers. Let dp[i] be the best damage using powers through index i. If taking powers[i], add its aggregated reward to the best dp[j] at the last index whose power is compatible; otherwise carry dp[i-1]. Find j by a moving pointer or binary search.

State the dependency

The key insight is that choice is about values, not original positions. Confirm the exact forbidden difference from the statement, because that defines compatibility. Use 64-bit totals. A moving pointer makes the sorted DP O(u log u) including sorting and O(u) space for u unique powers; binary search gives the same asymptotic bound. Test repeated powers, a single value, and values exactly at the incompatibility boundary.