
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
In a convex quadrilateral  $ABCD$  diagonals intersect at  $E$  and  $BE = \sqrt{2}\cdot ED, \: \angle BEC = 45^{\circ}.$   Let  $F$  be the foot of the perpendicular from  $A$  to  $BC$  and  $P$  be the second intersection point of the circumcircle of triangle  $BFD$  and line segment  $DC$ . Find  $\angle APD$ .

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