Large Numbers Don't Sway Me!

\[\large {30x \equiv 877 \pmod p \\ 24x \equiv 107 \pmod p}\]

Let \(x\) and \(p\) be positive integers satisfying the congruences above.

It is also given that \(0 \leq x < p\) and \(p\) is a 3-digit prime, find the value of \(x+p\).


This problem is adapted from the GCHQ Sample Maths Aptitude Test.

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