
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
Rectangle  $EFGH$  with side lengths  $8$ ,  $9$  lies inside rectangle  $ABCD$  with side lengths  $13$ ,  $14$ , with their corresponding sides parallel. Let  $\ell_A, \ell_B, \ell_C, \ell_D$  be the lines connecting  $A,B,C,D$ , respectively, with the vertex of  $EFGH$  closest to them. Let  $P = \ell_A \cap \ell_B$ ,  $Q = \ell_B \cap \ell_C$ ,  $R = \ell_C \cap \ell_D$ , and  $S = \ell_D \cap \ell_A$ . Suppose that the greatest possible area of quadrilateral  $PQRS$  is  $\frac{m}{n}$ , for relatively prime positive integers  $m$  and  $n$ . Find  $100m+n$ .

*Proposed by Yannick Yao*

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