Let the line $$\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z-3}{-4}$$ intersect the plane containing the lines $$\frac{x-4}{1}=\frac{y+1}{-2}=\frac{z}{1}$$ and $$4 a x-y+5 z-7 a=0=2 x-5 y-z-3, a \in \mathbb{R}$$ at the point $$P(\alpha, \beta, \gamma)$$. Then the value of $$\alpha+\beta+\gamma$$ equals _____________.

An ellipse $$E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ passes through the vertices of the hyperbola $$H: \frac{x^{2}}{49}-\frac{y^{2}}{64}=-1$$. Let the major and minor axes of the ellipse $$E$$ coincide with the transverse and conjugate axes of the hyperbola $$H$$, respectively. Let the product of the eccentricities of $$E$$ and $$H$$ be $$\frac{1}{2}$$. If $$l$$ is the length of the latus rectum of the ellipse $$E$$, then the value of $$113 l$$ is equal to _____________.

Let $$y=y(x)$$ be the solution curve of the differential equation

$$\sin \left( {2{x^2}} \right){\log _e}\left( {\tan {x^2}} \right)dy + \left( {4xy - 4\sqrt 2 x\sin \left( {{x^2} - {\pi \over 4}} \right)} \right)dx = 0$$, $$0 < x < \sqrt {{\pi \over 2}} $$, which passes through the point $$\left(\sqrt{\frac{\pi}{6}}, 1\right)$$. Then $$\left|y\left(\sqrt{\frac{\pi}{3}}\right)\right|$$ is equal to ______________.

Let $$M$$ and $$N$$ be the number of points on the curve $$y^{5}-9 x y+2 x=0$$, where the tangents to the curve are parallel to $$x$$-axis and $$y$$-axis, respectively. Then the value of $$M+N$$ equals ___________.