
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 square of side length 1. \(P\) and \(Q\) are two points on the plane such that \(Q\) is the circumcentre of \(\triangle BPC\) and \(D\) is the circumcentre of \(\triangle PQA\). Find the largest possible value of \(PQ^2\). Express the answer in the form \(a + \sqrt{b}\) or \(a - \sqrt{b}\), where \(a\) and \(b\) are rational numbers.

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