2024 Fermat Problem 25

A sequence of 11 positive real numbers, a_1, a_2, a_3, \ldots, a_{11}, satisfies a_1 = 4 and a_{11} = 1024 and a_n + a_{n-1} = \frac{5}{2}\sqrt{a_n \cdot a_{n-1}} for every integer n with 2 \le n \le 11. For example when n = 7, a_7 + a_6 = \frac{5}{2}\sqrt{a_7 \cdot a_6}. There are S such sequences. What are the rightmost two digits of S?

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