
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
A billiard table is shaped as a regular hexagon \( A B C D E F \). A ball is struck from the midpoint \( P \) of \( A B \), hits a point \( Q \) on the side \( B C \), and subsequently hits the sides \( C D \), \( D E \), \( E F \), and \( F A \) in sequence, finally hitting a certain point on the side \( A B \). Given that \(\angle B P Q = \theta\), determine the range of values for \(\theta\).

Hint: Use the principle that the angle of incidence is equal to the angle of reflection.

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