
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
Determine the values of \(p\) and \(q\) such that the triangle with sides \(p\), \(p+q\), and \(p+2q\) not only is a right triangle but also has its longest side not exceeding 12.
A) \(p = \frac{1+\sqrt{7}}{2}, q=1\)
B) \(p = 3, q = 1\)
C) \(p = 4, q = 1\)
D) \(p = 4, q = 2\)

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