
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
Hong and Song each have a shuffled deck of eight cards, four red and four black. Every turn, each player places down the two topmost cards of their decks. A player can thus play one of three pairs: two black cards, two red cards, or one of each color. The probability that Hong and Song play exactly the same pairs as each other for all four turns is  $\frac{m}{n}$ , where  $m$  and  $n$  are positive integers with  $\gcd(m,n)=1$ . Compute  $100m+n$ .

*Proposed by Sean Li*

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