Non-decreasing Sequence Averages

Let $S = a_1 + a_2 + \dots + a_N = 300N$. Replacing $a_1$ with $6a_1$ increases the sum by $5a_1$. New sum = $300N + 5a_1 = 400N \implies 5a_1 = 100N \implies a_1 = 20N$. Since the sequence is non-decreasing: $a_1 \le a_2 \le \dots \le a_N$. Sum $S = a_1 + a_2 + \dots + a_N \ge N a_1$. Also $S = 300N$, so $300N \ge N a_1 \implies a_1 \le 300$. Since $a_1 = 20N \le 300$, we have $N \le 15$. Also, $a_1 + (N-1)a_N \ge S$, and maximum $a_N$ can be anything, but for non-decreasing sequence, $a_1$ must be non-negative integer if we consider integers, or let's check bounds on $N$. $N$ can be any integer from 2 to 15 (if $N=1$, $a_1 = 20(1) = 20 \neq 300$). So $N \in \{2, 3, \dots, 15\}$, which gives 14 possible values for $N$, hence 14 possible values for $a_1 = 20N$.