Given a string of digits num and a target, insert '+', '-', and '*' between some of the digits (grouping the rest into multi-digit numbers, never with a leading zero) so the resulting expression, evaluated with standard operator precedence, equals target. Return every distinct expression that works. Building the expression as a plain string and parsing it from scratch once it's complete works, but re-derives the entire computation — digit by digit, operator by operator — using information the search already had the moment each piece was placed. Carrying the running total (and the value of the most recent term, needed to handle '*' correctly) forward as the expression is built means every finished expression already knows its own value; checking it against the target is one comparison, not a second pass over the string.