
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
\(AC\) and \(BD\) are two perpendicular chords in a circle. The chords intersect at \(E\), such that \(BE = 3\), \(EC = 2\), and \(AE = 6\). The exact perimeter of the quadrilateral \(ABCD\) may be written in the form \(m \sqrt{n} + p \sqrt{q}\), where \(m, n, p\), and \(q\) are positive integers, \(q > n\), and neither \(q\) nor \(n\) has a divisor that is a perfect square greater than 1. Determine the value of \(\sqrt{m n} + \sqrt{p + q}\).

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