Let $z_1, z_2$, and $z_3$ be complex numbers satisfying the following conditions

$$ 2=\left|2 z_1\right|=\left|z_2-1\right|=\left|z_3+1\right|=\left|\frac{1}{z_1}+\frac{1}{z_2-1}+\frac{1}{z_3+1}\right| . $$

What is the value of $\left|4 z_1+z_2+z_3\right|$ ?

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x)=\left|x^3-3 x\right|[x]$, where $[x]$ denotes the greatest integer less than or equal to $x$. Which one of the following statements is TRUE?

Let $\ell$ be the tangent line to the ellipse $x^2+16 y^2=4$ at $\left(1, \frac{\sqrt{3}}{4}\right)$. What is the equation of the line perpendicular to $\ell$ passing through $(2,0)$ ?

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}+\vec{b}|=15$ and

$$ \vec{a} \times(3 \hat{i}-4 \hat{j}+5 \hat{k})=(3 \hat{i}-4 \hat{j}+5 \hat{k}) \times \vec{b} $$

What is the value of $|(\vec{a}+\vec{b}) \cdot(2 \hat{i}+3 \hat{j}+\hat{k})|$ ?