
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 sufficient and necessary condition for the system of equations
$$
\begin{array}{c}
x^{2}+y^{2}+z^{2}=(B a)^{2} \\
x^{2}\left(A x^{2}+B y^{2}\right)+y^{2}\left(A y^{2}+B z^{2}\right)+z^{2}\left(A z^{2}+B x^{2}\right)= \\
\frac{1}{4}(2 A+B)(B a)^{4}
\end{array}
$$
to have positive integer solutions. Express the condition in terms of a relation between \( A \) and \( B \), and provide a proof.

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