
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

For a prime number \( p \), define the sets
$$
\begin{aligned}
S_{1}(p) = & \left\{(a, b, c) \in \mathbf{Z}^{3} \mid a^{2} b^{2} + b^{2} c^{2} + c^{2} a^{2} + 1 \equiv 0 \pmod{p}\right\}
\end{aligned}
$$
and
$$
S_{2}(p) = \left\{(a, b, c) \in \mathbf{Z}^{3} \mid a^{2} b^{2} c^{2} \left( a^{2} b^{2} c^{2} + a^{2} + b^{2} + c^{2} \right) \equiv 0 \pmod{p}\right\}.
$$
Find all prime numbers \( p \) such that
$$
S_{1}(p) \subseteq S_{2}(p).
$$

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