Consider the region R = {(x, y) $$\in$$ R $$\times$$ R : x $$\ge$$ 0 and y² $$\le$$ 4 $$-$$ x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the circles in F. Let ($$\alpha$$, $$\beta$$) be a point where the circle C meets the curve y² = 4 $$-$$ x.

The value of $$\alpha$$ is ___________.

Let f₁ : (0, $$\infty$$) $$\to$$ R and f₂ : (0, $$\infty$$) $$\to$$ R be defined by $${f_1}(x) = \int\limits_0^x {\prod\limits_{j = 1}^{21} {{{(t - j)}^j}dt} } $$, x > 0 and $${f_2}(x) = 98{(x - 1)^{50}} - 600{(x - 1)^{49}} + 2450,x > 0$$, where, for any positive integer n and real numbers a₁, a₂, ....., a_n, $$\prod\nolimits_{i = 1}^n {{a_i}} $$ denotes the product of a₁, a₂, ....., a_n. Let m_i and n_i, respectively, denote the number of points of local minima and the number of points of local maxima of function f_i, i = 1, 2 in the interval (0, $$\infty$$).

The value of $$2{m_1} + 3{n_1} + {m_1}{n_1}$$ is ___________.

Let f₁ : (0, $$\infty$$) $$\to$$ R and f₂ : (0, $$\infty$$) $$\to$$ R be defined by $${f_1}(x) = \int\limits_0^x {\prod\limits_{j = 1}^{21} {{{(t - j)}^j}dt} } $$, x > 0 and $${f_2}(x) = 98{(x - 1)^{50}} - 600{(x - 1)^{49}} + 2450,x > 0$$, where, for any positive integer n and real numbers a₁, a₂, ....., a_n, $$\prod\nolimits_{i = 1}^n {{a_i}} $$ denotes the product of a₁, a₂, ....., a_n. Let m_i and n_i, respectively, denote the number of points of local minima and the number of points of local maxima of function f_i, i = 1, 2 in the interval (0, $$\infty$$).

The value of $$6{m_2} + 4{n_2} + 8{m_2}{n_2}$$ is ___________.

Let $${g_i}:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R,i = 1,2$$, and $$f:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R$$ be functions such that $${g_1}(x) = 1,{g_2}(x) = |4x - \pi |$$ and $$f(x) = {\sin ^2}x$$, for all $$x \in \left[ {{\pi \over 8},{{3\pi } \over 8}} \right]$$. Define $${S_i} = \int\limits_{{\pi \over 8}}^{{{3\pi } \over 8}} {f(x).{g_i}(x)dx} $$, i = 1, 2

The value of $${{16{S_1}} \over \pi }$$ is _____________.