
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  $ABCD$  be a square of side length  $6$ .  Points  $E$  and  $F$  are selected on rays  $AB$  and  $AD$  such that segments  $EF$  and  $BC$  intersect at a point  $L$ ,  $D$  lies between  $A$  and  $F$ , and the area of  $\triangle AEF$  is 36. Clio constructs triangle  $PQR$  with  $PQ=BL$ ,  $QR=CL$  and  $RP=DF$ , and notices that the area of  $\triangle PQR$  is  $\sqrt{6}$ . If the sum of all possible values of  $DF$  is  $\sqrt{m} + \sqrt{n}$  for positive integers  $m \ge n$ , compute  $100m+n$ .

*Based on a proposal by Calvin Lee*

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