
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
Points  $A$ ,  $B$ ,  $C$ ,  $D$ , and  $E$  are on the same plane such that  $A,E,C$  lie on a line in that order,  $B,E,D$  lie on a line in that order,  $AE = 1$ ,  $BE = 4$ ,  $CE = 3$ ,  $DE = 2$ , and  $\angle AEB = 60^\circ$ . Let  $AB$  and  $CD$  intersect at  $P$ . The square of the area of quadrilateral  $PAED$  can be expressed as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. What is  $m+n$ ?


*2021 CCA Math Bonanza Individual Round #9*

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