
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
Let \(ABCD\) be a quadrilateral inscribed in a circle with center \(O\). Let \(P\) denote the intersection of \(AC\) and \(BD\). Let \(M\) and \(N\) denote the midpoints of \(AD\) and \(BC\). If \(AP=1\), \(BP=3\), \(DP=\sqrt{3}\), and \(AC\) is perpendicular to \(BD\), find the area of triangle \(MON\).

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