
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
If $a, b, c$ are non-negative integers less than $7$ such that 
\begin{align*}
a + 2b + 3c &\equiv 0 \pmod{7}, \\
2a + 3b + c &\equiv 2 \pmod{7}, \\
3a + b + 2c &\equiv 4 \pmod{7},
\end{align*}
determine the remainder when $abc$ is divided by $7$.

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