Suppose \(x\) is an integer that satisfies the following system of congruences:
\[
\begin{align*}
x &\equiv 4 \pmod{5}, \\
x &\equiv 3 \pmod{7}, \\
x &\equiv 2 \pmod{11}.
\end{align*}
What is the smallest positive integer \(x\) that satisfies all three congruences? Let's think step by step and output the final answer within \boxed{}.
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