Let \(h(x) \) denote a monic \(8^\text{th} \)-degree polynomial such that \(h(m) = \binom m6 \) for \(m = 6,7,8,\ldots,13\).

Find \(h(5) \).

**Notation**: \( \binom mn \) denotes the binomial coefficient \( \binom mn = \frac{m!}{n!(m-n)!} \) for non-negative integers \(m\geq n \).

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