The Broken Balance

Rules you need

• Original variant "pair subtraction". Each component is a pair of heaps A and B. The pairs remain independent.
• At each turn, choose exactly ONE pair, then remove 1, 2 or 3 chips from A alone, OR B alone, OR exactly the same number (1, 2 or 3) of both piles. No pile can become negative. Never remove two different pairs in the same round.
• The two sides alternate. The player without a legal blow loses: taking the last chips makes it win. There is neither pass nor rule misery. The tokens of the same pile are indistinguishable; the pairs and piles A/B are labeled.

The puzzle

You play first. To analyze alone, calculate the winning and losing positions of both sides.

Gives a complete list of the winners and a general proof of answer: only one cooperative suite is not enough.