
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 \( ABC \) be an acute triangle, and let \( M \) be the midpoint of \( AC \). A circle \(\omega\) passing through \( B \) and \( M \) meets the sides \( AB \) and \( BC \) again at \( P \) and \( Q \), respectively. Let \( T \) be the point such that the quadrilateral \( BPTQ \) is a parallelogram. Suppose that \( T \) lies on the circumcircle of the triangle \( ABC \). Determine all possible values of \( BT / BM \).

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