
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 $a, b, c$ be non-negative integers less than $5$ satisfying the equations:
\begin{align*}
a + 2b + 3c &\equiv 3 \pmod{5}, \\
2a + 3b + c &\equiv 2 \pmod{5}, \\
3a + b + 2c &\equiv 1 \pmod{5}.
\end{align*}
Find the remainder when $abc$ is divided by $5$.

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