
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 that $x$ is an integer that satisfies the following congruences:
\begin{align*}
2+x &\equiv 2^2 \pmod{5^3} \\
3+x &\equiv 3^2 \pmod{7^3} \\
4+x &\equiv 5^2 \pmod{11^3} 
\end{align*}
What is the remainder when $x$ is divided by $385$ (where $385 = 5 \cdot 7 \cdot 11$)?

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