
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
In the game of Connect Four, there are seven vertical columns which have spaces for six tokens, forming a \(7 \times 6\) grid of spaces. Two players, White and Black, move alternately. A player takes a turn by picking a column that is not already full and dropping a token of their color into the lowest unoccupied space in that column. The game ends when one player has four consecutive tokens of their color in a line: either horizontally, vertically, or diagonally. The player who achieves this wins the game.

Assume that the two players play this game randomly. Each player, on their turn, picks a random column that is not full and drops a token of their color into that column. This process continues until one player wins or all columns are filled. Let \( P \) be the probability that all columns are filled without any player obtaining four tokens in a row of their color. Estimate \( P \). 

An estimate of \( E > 0 \) earns \( \lfloor 20 \min (P / E, E / P)\rfloor \) points.

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