Given the function $f(x) = \sin^2 x + \sin x + \cos^2 x$, we will address both parts of the problem sequentially.

Part (1) Find the Symmetry Center of the Function $f(x)$

First, simplify the function $f(x)$:

$f(x) = \sin^2 x + \sin x + \cos^2 x$

Using the Pythagorean identity, $\sin^2 x + \cos^2 x = 1$:

$f(x) = 1 + \sin x$

This function is linear in terms of $\sin x$, and we can analyze its symmetry properties. To check if it has any symmetry center, we