
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
Suppose $x, y, z$ are integers from the set of positive integers less than $11$ such that 
\[
xyz \equiv 3 \pmod{11}, \\
7z \equiv 4 \pmod{11}, \\
9y \equiv 5 + y \pmod{11}.
\]
Determine the remainder when $x+y+z$ is divided by $11$.

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