Putnam 2024 B6

For a real number $a$, let $F_a(x) = \sum_{n \geq 1} n^a e^{2n} x^{n^2}$ for $0 \leq x < 1$. Find a real number $c$ such that

$$\begin{aligned}& \lim_{x \to 1^-} F_a(x) e^{-1/(1-x)} = 0 \text{for all $a < c$, and} \\ & \lim_{x \to 1^-} F_a(x) e^{-1/(1-x)} = \infty \text{for all $a > c$.}\end{aligned}$$