Let $\mathbb{R}$ denote the set of all real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow(0,4)$ be functions defined by

$$ f(x)=\log _e\left(x^2+2 x+4\right), \text { and } g(x)=\frac{4}{1+e^{-2 x}} $$

Define the composite function $f \circ g^{-1}$ by $\left(f \circ g^{-1}\right)(x)=f\left(g^{-1}(x)\right)$, where $g^{-1}$ is the inverse of the function $g$.

Then the value of the derivative of the composite function $f \circ g^{-1}$ at $x=2$ is ________________.

Let ℝ denote the set of all real numbers. Let f: ℝ → ℝ be a function such that f(x) > 0 for all x ∈ ℝ, and f(x+y) = f(x)f(y) for all x, y ∈ ℝ.

Let the real numbers a₁, a₂, ..., a₅₀ be in an arithmetic progression. If f(a₃₁) = 64f(a₂₅), and

$ \sum\limits_{i=1}^{50} f(a_i) = 3(2^{25}+1), $

then the value of

$ \sum\limits_{i=6}^{30} f(a_i) $

is ________________.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x, y \in \mathbb{R}$, and $g: \mathbb{R} \rightarrow(0, \infty)$ be a function such that $g(x+y)=g(x) g(y)$ for all $x, y \in \mathbb{R}$. If $f\left(\frac{-3}{5}\right)=12$ and $g\left(\frac{-1}{3}\right)=2$, then the value of $\left(f\left(\frac{1}{4}\right)+g(-2)-8\right) g(0)$ is _________.

Let the function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by

$$ f(x)=\frac{\sin x}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}+\frac{2}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)} . $$

Then the number of solutions of $f(x)=0$ in $\mathbb{R}$ is _________.