
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

Given a convex quadrilateral ABCD, \( P \) is a point inside ABCD such that the angle \( PXY \) is acute for any two adjacent vertices \( X \) and \( Y \) of ABCD. Define \( Q_{n+1} \) to be derived from \( Q_n \) by reflecting \( P \) in each side of \( Q_n \) to obtain the vertices of \( Q_{n+1} \). 

1. Which of \( Q_1, Q_2, \ldots, Q_{12} \) must be similar to \( Q_{1997} \)?
2. If \( Q_{1997} \) is cyclic, which of \( Q_1, Q_2, \ldots, Q_{12} \) must be cyclic?

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