Let's analyze and solve the given problem step-by-step.

Problem Statement:

Given the function $f(x) = \sin^2(x) + \sin(x) + \cos^2(x)$:

1. Find the center of symmetry of the function $f(x)$.
2. Determine the range of the real number $n$ if the function $f(x)$ has exactly 2024 zeros in the interval $\left[0, \frac{n\pi}{2}\right]$.

Part 1: Finding the Center of Symmetry

First, we simplify the given function:

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

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

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

Next, we need to check if the function ( f(x) =