The Limited Operator Chain

Given a string of digits and a target integer, find the number of ways to insert '+' or '-' between every digit to reach the target. Constraint: You cannot use the same operator more than twice in a row (e.g., '1+2+3+4' is invalid because there are three '+' in a sequence). Input: A string of digits and a target integer separated by space. Output: The count of valid expressions. Example: Input: 111 1 Expressions: 1+1-1=1, 1-1+1=1. Both are valid. (1+1+1 is invalid as it uses '+' twice but the logic allows up to 2, wait—2 same ops is the limit, so +++ is the first invalid case. 1+1+1 is actually valid, 1+1+1+1 is not).