
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
Let \( P \) be a point selected uniformly at random in the cube \([0,1]^{3}\). The plane parallel to \( x+y+z=0 \) passing through \( P \) intersects the cube in a two-dimensional region \(\mathcal{R}\). Let \( t \) be the expected value of the perimeter of \(\mathcal{R}\). If \( t^{2} \) can be written as \( \frac{a}{b} \), where \( a \) and \( b \) are relatively prime positive integers, compute \( 100a + b \).

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