
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 \( a \) be a given positive integer, and \( A \) and \( B \) be two real numbers. Determine the necessary and sufficient conditions (expressed in terms of \( A \) and \( B \)) for the system of equations
$$
\left\{\begin{array}{l}
x^{2}+y^{2}+z^{2}=(13 a)^{2} \\
x^{2}\left(A x^{2}+B y^{2}\right)+y^{2}\left(A y^{2}+B z^{2}\right)+z^{2}\left(A z^{2}+B x^{2}\right)=\frac{1}{4}(2 A+B)(13 a)^{4}
\end{array}\right.
$$
to have positive integer solutions. Provide a proof for these conditions.

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