Let the domain of the function $f(x)=\cos ^{-1}\left(\frac{4 x+5}{3 x-7}\right)$ be $[\alpha, \beta]$ and the domain of $g(x)=\log _2\left(2-6 \log _{27}(2 x+5)\right)$ be $(\gamma, \delta)$.

Then $|7(\alpha+\beta)+4(\gamma+\delta)|$ is equal to ______________.

Let $$A=\{(x, y): 2 x+3 y=23, x, y \in \mathbb{N}\}$$ and $$B=\{x:(x, y) \in A\}$$. Then the number of one-one functions from $$A$$ to $$B$$ is equal to _________.

If a function $$f$$ satisfies $$f(\mathrm{~m}+\mathrm{n})=f(\mathrm{~m})+f(\mathrm{n})$$ for all $$\mathrm{m}, \mathrm{n} \in \mathbf{N}$$ and $$f(1)=1$$, then the largest natural number $$\lambda$$ such that $$\sum_\limits{\mathrm{k}=1}^{2022} f(\lambda+\mathrm{k}) \leq(2022)^2$$ is equal to _________.

If the range of $$f(\theta)=\frac{\sin ^4 \theta+3 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta}, \theta \in \mathbb{R}$$ is $$[\alpha, \beta]$$, then the sum of the infinite G.P., whose first term is 64 and the common ratio is $$\frac{\alpha}{\beta}$$, is equal to __________.