
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
Let  $ABC$  be a triangle with circumcircle  $\omega$  and circumcenter  $O.$  Suppose that  $AB = 15$ ,  $AC = 14$ , and  $P$  is a point in the interior of  $\triangle ABC$  such that  $AP = \frac{13}{2}$ ,  $BP^2 = \frac{409}{4}$ , and  $P$  is closer to  $\overline{AC}$  than to  $\overline{AB}$ . Let  $E$ ,  $F$  be the points where  $\overline{BP}$ ,  $\overline{CP}$  intersect  $\omega$  again, and let  $Q$  be the intersection of  $\overline{EF}$  with the tangent to  $\omega$  at  $A.$  Given that  $AQOP$  is cyclic and that  $CP^2$  is expressible in the form  $\frac{a}{b} - c \sqrt{d}$  for positive integers  $a$ ,  $b$ ,  $c$ ,  $d$  such that  $\gcd(a, b) = 1$  and  $d$  is not divisible by the square of any prime, compute  $1000a+100b+10c+d.$  

*Proposed by Edward Wan*

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