SIMON-MARAIS 2017 A2

Let $a_1, a_2, a_3, \ldots$ be the sequence of real numbers defined by $a_1 = 1$ and

$$a_m = \frac{1}{a_1^2 + a_2^2 + \cdots + a_{m-1}^2}$$

for $m \geq 2$.

Determine whether there exists a positive integer $N$ such that

$$a_1 + a_2 + \cdots + a_N > 2017^{2017}$$.