
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]


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Acute triangle \( ABC \) has circumcircle \(\Gamma\). Let \( M \) be the midpoint of \( BC \). Points \( P \) and \( Q \) lie on \(\Gamma\) such that \(\angle A P M = 90^\circ\) and \( Q \neq A \) lies on line \( AM \). Segments \( PQ \) and \( BC \) intersect at \( S \). Suppose that \( BS = 1 \), \( CS = 3 \), \( PQ = 8 \sqrt{\frac{7}{37}} \), and the radius of \(\Gamma\) is \( r \). If the sum of all possible values of \( r^{2} \) can be expressed as \(\frac{a}{b}\) for relatively prime positive integers \( a \) and \( b \), compute \( 100a + b \).

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