For, $n = 1, \dots, 365$, why does: $\frac{1}{n} \sum_{k=0}^{n-1}(1 - k/365)^n \leq \left(\frac{1}{n}\int_0^n (1 - x/365)dx\right)^n$

Question

From Jensen's I get the slightly larger bound of

$$\frac{1}{n} \sum_{k=0}^{n-1}\left(1 - \frac{k}{365}\right)^n < \left(\frac{1}{n} \sum_{k=0}^{n-1}\left(1 - \frac{k}{365}\right)\right)^n.$$

But I'm blanking on why

$$ \frac{1}{n} \sum_{k=0}^{n-1}\left(1 - \frac{k}{365}\right)^n < \left(\frac{1}{n}\int_0^n \left(1 - \frac{x}{365}\right)dx\right)^n < \left(\frac{1}{n} \sum_{k=0}^{n-1}\left(1 - \frac{k}{365}\right)\right)^n,$$

for $n \in \{1, \dots, 365\}$.