If the value of \[\sin^{24}\frac{\pi}{24} + \cos^{24}\frac{\pi}{24}\]

is expressed in simplest form as \( \dfrac{a + b\sqrt{c}}{d},\) find the last three digits of \(a + b + c + d.\)

**Note:** Simplest form refers to the condition that \(a,b,c,d\) are positive integers with \(\gcd(a,b,d)=1\) and \(c\) is not divisible by the square of any prime.

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