Consider the vectors

$$ \vec{x}=\hat{\imath}+2 \hat{\jmath}+3 \hat{k}, \quad \vec{y}=2 \hat{\imath}+3 \hat{\jmath}+\hat{k}, \quad \text { and } \quad \vec{z}=3 \hat{\imath}+\hat{\jmath}+2 \hat{k} $$

For two distinct positive real numbers $\alpha$ and $\beta$, define

$$ \vec{X}=\alpha \vec{x}+\beta \vec{y}-\vec{z}, \quad \vec{Y}=\alpha \vec{y}+\beta \vec{z}-\vec{x}, \quad \text { and } \quad \vec{Z}=\alpha \vec{z}+\beta \vec{x}-\vec{y} . $$

If the vectors $\vec{X}, \vec{Y}$, and $\vec{Z}$ lie in a plane, then the value of $\alpha+\beta-3$ is ____________.

For a non-zero complex number $z$, let $\arg (z)$ denote the principal argument of $z$, with $-\pi<\arg (z) \leq \pi$. Let $\omega$ be the cube root of unity for which $0<\arg (\omega)<\pi$. Let

$$ \alpha=\arg \left(\sum\limits_{n=1}^{2025}(-\omega)^n\right) $$

Then the value of $\frac{3 \alpha}{\pi}$ is ________________.

Let $\mathbb{R}$ denote the set of all real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow(0,4)$ be functions defined by

$$ f(x)=\log _e\left(x^2+2 x+4\right), \text { and } g(x)=\frac{4}{1+e^{-2 x}} $$

Define the composite function $f \circ g^{-1}$ by $\left(f \circ g^{-1}\right)(x)=f\left(g^{-1}(x)\right)$, where $g^{-1}$ is the inverse of the function $g$.

Then the value of the derivative of the composite function $f \circ g^{-1}$ at $x=2$ is ________________.

Let

$$ \alpha=\frac{1}{\sin 60^{\circ} \sin 61^{\circ}}+\frac{1}{\sin 62^{\circ} \sin 63^{\circ}}+\cdots+\frac{1}{\sin 118^{\circ} \sin 119^{\circ}} $$

Then the value of

$$ \left(\frac{\operatorname{cosec} 1^{\circ}}{\alpha}\right)^2 $$

is _____________.