TSTST 2025 Problem 7

For a positive real number $c$, the sequence $a_1, a_2, \dots$ of real numbers is defined as follows. Let $a_1 = c$, and for $n \geq 2$, let

$$a_n = \sum_{i=1}^{n-1} (a_i)^{n-i+1}.$$

Find all positive real numbers $c$ such that $a_i > a_{i+1}$ for all positive integers $i$.