
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 unit square and let \(P\) and \(Q\) be points in the plane such that \(Q\) is the circumcenter of triangle \(BPC\) and \(D\) is the circumcenter of triangle \(PQA\). Find all possible values of the length of segment \(PQ\).

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