
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
Pairwise distinct points  $P_1,P_2,\ldots, P_{16}$  lie on the perimeter of a square with side length  $4$  centered at  $O$  such that  $\lvert P_iP_{i+1} \rvert = 1$  for  $i=1,2,\ldots, 16$ . (We take  $P_{17}$  to be the point  $P_1$ .) We construct points  $Q_1,Q_2,\ldots,Q_{16}$  as follows: for each  $i$ , a fair coin is flipped. If it lands heads, we define  $Q_i$  to be  $P_i$ ; otherwise, we define  $Q_i$  to be the reflection of  $P_i$  over  $O$ . (So, it is possible for some of the  $Q_i$  to coincide.)  Let  $D$  be the length of the vector  $\overrightarrow{OQ_1} + \overrightarrow{OQ_2} + \cdots + \overrightarrow{OQ_{16}}$ . Compute the expected value of  $D^2$ .

*Ray Li*

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