Daniel has a weighted coin that flips heads \(\frac{2}{5}\) of the time and tails \(\frac{3}{5}\) of the time. If he flips it \(9\) times, the probability that it will show heads exactly \(n\) times is greater than or equal to the probability that it will show heads exactly \(k\) times, for all \(k=0, 1,\dots, 9, k\ne n\).

If the probability that the coin will show heads exactly \(n\) times in \(9\) flips is \(\frac{p}{q}\) for positive coprime integers \(p\) and \(q\), then find the last three digits of \(p\).

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