Senior Kangaroo 2024 Q18

Zarah has a bag containing exactly four blue counters and exactly four red counters, and a large supply, outside the bag, of extra blue and red counters. She may, on any given $\text{move}$, do one of four different things:

• Remove a blue counter from the bag;
• Remove a red counter from the bag;
• Add a blue counter to the bag;
• Add a red counter to the bag.

At each move, she selects one of those four options at random and with equal probability. The probability that after four consecutive moves, Zarah's bag again contains exactly four blue counters and exactly four red counters is $\frac{V}{W}$ when written in its lowest terms. What is the value of $10V + W$?