
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
The quadrilateral  $ABCD$  is inscribed in the parabola  $y=x^2$ . It is known that angle  $BAD=90$ , the dioganal  $AC$  is parallel to the axis  $Ox$  and  $AC$  is the bisector of the angle BAD.
Find the area of the quadrilateral  $ABCD$  if the length of the dioganal  $BD$  is equal to  $p$ .

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