
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 \( O = (0,0) \), \( Q = (13,4) \), \( A = (a, a) \), \( B = (b, 0) \), where \(a\) and \(b\) are positive real numbers with \(b \geq a\). The point \( Q \) is on the line segment \( A B \).

(a) Determine the values of \( a \) and \( b \) for which \( Q \) is the midpoint of \( A B \).

(b) Determine all values of \( a \) and \( b \) for which \( Q \) is on the line segment \( A B \) and the triangle \( O A B \) is isosceles and right-angled.

(c) There are infinitely many line segments \( A B \) that contain the point \( Q \). For how many of these line segments are \( a \) and \( b \) both integers?

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