
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 rectangle  $ABCD$  has side lengths  $AB=6 \text{ miles}$  and  $BC=9\text{ miles}.$  A pigeon hovers at point  $P$ , which is 5 miles above some randomly chosen point inside  $ABCD$ . Given that the expected value of \[AP^2+CP^2-BP^2-DP^2\] can be expressed as  $\tfrac{a}{b}$ , what is  $ab$ ? 

*2022 CCA Math Bonanza Lightning Round 2.2*

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