
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 trapezoid \(ABCD\), diagonal \(AC\) is perpendicular to the side \(CD\) and diagonal \(DB\) is perpendicular to the side \(AB\). The extensions of the sides \(AB\) and \(DC\) intersect at point \(K\), forming triangle \(AKD\) with an angle of \(45^\circ\) at vertex \(K\). The area of trapezoid \(ABCD\) is \(P\). Find the area of triangle \(AKD\).

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