
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 1 \pmod{7}, \\
2a+3b+c &\equiv 2 \pmod{7}, \\
3a+b+2c &\equiv 1 \pmod{7},
\end{align*}
then determine the remainder when $abc$ is divided by $7$.

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