Minimum Operations to Equalize Binary String

Given a binary string s and an integer k, each operation flips exactly k distinct bits; find the minimum operations to make all characters '1' (or -1 if impossible). The core challenge is deciding reachability and shortest sequence of fixed-size flips under parity/combinatorial constraints, reducible to a shortest-path/DP problem on the space of zero-counts or to solving linear constraints rather than brute-force 2^n search.