If $$\lim _\limits{x \rightarrow 0} \frac{a x^2 e^x-b \log _e(1+x)+c x e^{-x}}{x^2 \sin x}=1$$, then $$16\left(a^2+b^2+c^2\right)$$ is equal to ________.

Let $$A=\{1,2,3, \ldots \ldots \ldots \ldots, 100\}$$. Let $$R$$ be a relation on $$\mathrm{A}$$ defined by $$(x, y) \in R$$ if and only if $$2 x=3 y$$. Let $$R_1$$ be a symmetric relation on $$A$$ such that $$R \subset R_1$$ and the number of elements in $$R_1$$ is $$\mathrm{n}$$. Then, the minimum value of $$\mathrm{n}$$ is _________.

Let the coefficient of $$x^r$$ in the expansion of $$(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^2+\ldots \ldots \ldots .+(x+2)^{n-1}$$ be $$\alpha_r$$. If $$\sum_\limits{r=0}^n \alpha_r=\beta^n-\gamma^n, \beta, \gamma \in \mathbb{N}$$, then the value of $$\beta^2+\gamma^2$$ equals _________.

Let A be a $$3 \times 3$$ matrix and $$\operatorname{det}(A)=2$$. If $$n=\operatorname{det}(\underbrace{\operatorname{adj}(\operatorname{adj}(\ldots . .(\operatorname{adj} A))}_{2024-\text { times }}))$$, then the remainder when $$n$$ is divided by 9 is equal to __________.