
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}
user