
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
A convex quadrilateral \(ABCD\) has an area of \(2601\) and contains a point \(P\) such that \(PA = 25\), \(PB = 35\), \(PC = 30\), and \(PD = 50\). Given that the diagonals of \(ABCD\) intersect each other at right angles at point \(P\), find the perimeter of quadrilateral \(ABCD\).
A) \(\sqrt{1850} + \sqrt{2125} + \sqrt{3400} + \sqrt{3125}\)
B) \(\sqrt{1800} + \sqrt{2100} + \sqrt{3500} + \sqrt{3100}\)
C) \(\sqrt{1900} + \sqrt{2200} + \sqrt{3300} + \sqrt{3200}\)
D) \(\sqrt{2000} + \sqrt{2300} + \sqrt{3400} + \sqrt{3000}\)

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