
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A and B play a game as follows. Each throws a dice. Suppose A gets \(x\) and B gets \(y\). If \(x\) and \(y\) have the same parity, then A wins. If not, they make a list of all two-digit numbers \(ab \leq xy\) with \(1 \leq a, b \leq 6\). Then they take turns (starting with A) replacing two numbers on the list by their non-negative difference. When just one number remains, it is compared to \(x\). If it has the same parity A wins, otherwise B wins. Find the probability that A wins.

Present the answer in LaTex format: \boxed{Your answer}
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