Let $$a_1, a_2, a_3, \ldots$$ be in an arithmetic progression of positive terms.

Let $$A_k=a_1^2-a_2^2+a_3^2-a_4^2+\ldots+a_{2 k-1}^2-a_{2 k}^2$$.

If $$\mathrm{A}_3=-153, \mathrm{~A}_5=-435$$ and $$\mathrm{a}_1^2+\mathrm{a}_2^2+\mathrm{a}_3^2=66$$, then $$\mathrm{a}_{17}-\mathrm{A}_7$$ is equal to ________.

Let $$\overrightarrow{\mathrm{a}}=\hat{i}-3 \hat{j}+7 \hat{k}, \overrightarrow{\mathrm{b}}=2 \hat{i}-\hat{j}+\hat{k}$$ and $$\overrightarrow{\mathrm{c}}$$ be a vector such that $$(\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}=3(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}})$$. If $$\vec{a} \cdot \vec{c}=130$$, then $$\vec{b} \cdot \vec{c}$$ is equal to __________.

If the constant term in the expansion of $$\left(1+2 x-3 x^3\right)\left(\frac{3}{2} x^2-\frac{1}{3 x}\right)^9$$ is $$\mathrm{p}$$, then $$108 \mathrm{p}$$ is equal to ________.

The number of distinct real roots of the equation $$|x||x+2|-5|x+1|-1=0$$ is __________.