If the area of the region $$\left\{(x, \mathrm{y}):\left|x^{2}-2\right| \leq y \leq x\right\}$$ is $$\mathrm{A}$$, then $$6 \mathrm{A}+16 \sqrt{2}$$ is equal to __________.

The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to __________.

Let $$\mathrm{S}$$ be the set of values of $$\lambda$$, for which the system of equations

$$6 \lambda x-3 y+3 z=4 \lambda^{2}$$,

$$2 x+6 \lambda y+4 z=1$$,

$$3 x+2 y+3 \lambda z=\lambda$$ has no solution. Then $$12 \sum_\limits{i \in S}|\lambda|$$ is equal to ___________.

Suppose $$a_{1}, a_{2}, 2, a_{3}, a_{4}$$ be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is $$\frac{49}{2}$$, then $$a_{4}$$ is equal to __________.