
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 from the set of positive integers less than $11$ such that:
1. $abc \equiv 3 \pmod{11}$,
2. $8c \equiv 5 \pmod{11}$,
3. $a + 3b \equiv 10 \pmod{11}$.

Find the remainder when $a+b+c$ is divided by $11$, and confirm if $a, b, c$ can also be three consecutive integers.

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