
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 symmedian point  $K$ , and let  $\theta = \angle AKB-90^{\circ}$ . Suppose that  $\theta$  is both positive and less than  $\angle C$ . Consider a point  $K'$  inside  $\triangle ABC$  such that  $A,K',K,$  and  $B$  are concyclic and  $\angle K'CB=\theta$ . Consider another point  $P$  inside  $\triangle ABC$  such that  $K'P\perp BC$  and  $\angle PCA=\theta$ . If  $\sin \angle APB = \sin^2 (C-\theta)$  and the product of the lengths of the  $A$ - and  $B$ -medians of  $\triangle ABC$  is  $\sqrt{\sqrt{5}+1}$ , then the maximum possible value of  $5AB^2-CA^2-CB^2$  can be expressed in the form  $m\sqrt{n}$  for positive integers  $m,n$  with  $n$  squarefree. Compute  $100m+n$ .

*Proposed by Vincent Huang*

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