
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 integers in the set $\{1, 2, 3, 4, 5, 6\}$ satisfying the conditions:
\[
abc \equiv 1 \pmod{7},
\]
\[
2c \equiv 5 \pmod{7},
\]
\[
3b \equiv 4 + b \pmod{7}.
\]
Determine the remainder when $a+b+c$ is divided by $7$.

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