
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 point \( P \) lies at the center of square \( ABCD \). A sequence of points \( \{P_n\} \) is determined by \( P_0 = P \), and given point \( P_i \), point \( P_{i+1} \) is obtained by reflecting \( P_i \) over one of the four lines \( AB, BC, CD, DA \), chosen uniformly at random and independently for each \( i \). What is the probability that \( P_8 = P \)?

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