
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 $a$ and two real numbers $A$ and $B$, determine the necessary and sufficient condition for the system of equations
$$
\begin{aligned}
&x^{2}+y^{2}+z^{2}=(13 a)^{2}, \\
&x^{2}\left(A x^{2}+B y^{2}\right)+y^{2}\left(A y^{2}+B z^{2}\right)+z^{2}\left(A z^{2}+B x^{2}\right) =\frac{1}{4}(2 A+B)(13 a)^{4}
\end{aligned}
$$
to have positive integer solutions (expressed in terms of the relationship between $A$ and $B$, along with a proof).

(from the 5th Chinese High School Mathematics Winter Camp, 1990)

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