The range of $a \in \mathbb{R}$ for which the function
$$
f(x)=(4 a-3)\left(x+\log _e 5\right)+2(a-7) \cot \left(\frac{x}{2}\right) \sin ^2\left(\frac{x}{2}\right)
$$
$x \neq 2 n \pi, n \in \mathbb{N}$, has critical points, is:
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