
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
Adam is playing Minesweeper on a  $9\times9$  grid of squares, where exactly  $\frac13$  (or 27) of the squares are mines (generated uniformly at random over all such boards). Every time he clicks on a square, it is either a mine, in which case he loses, or it shows a number saying how many of the (up to eight) adjacent squares are mines.

First, he clicks the square directly above the center square, which shows the number  $4$ . Next, he clicks the square directly below the center square, which shows the number  $1$ . What is the probability that the center square is a mine?

*Proposed by Adam Bertelli*

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