
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
Given a positive integer \( n \), for what real \( u, v \) do the simultaneous equations \( a^2 + b^2 + c^2 = 169n^2 \) and \( a^2(ua^2 + vb^2) + b^2(ub^2 + vc^2) + c^2(uc^2 + va^2) = \frac{(2u + v)(13n)^4}{4} \) have a solution in positive integers \( a, b, c \)?

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