Let ℕ denote the set of all natural numbers, and ℤ denote the set of all integers. Consider the functions f: ℕ → ℤ and g: ℤ → ℕ defined by

$$ f(n) = \begin{cases} \frac{(n + 1)}{2} & \text{if } n \text{ is odd,} \\ \frac{(4-n)}{2} & \text{if } n \text{ is even,} \end{cases} $$

and

$$ g(n) = \begin{cases} 3 + 2n & \text{if } n \ge 0 , \\ -2n & \text{if } n < 0 . \end{cases} $$

Define $$(g \circ f)(n) = g(f(n))$$ for all $n \in \mathbb{N}$, and $$(f \circ g)(n) = f(g(n))$$ for all $n \in \mathbb{Z}$.

Then which of the following statements is (are) TRUE?

Let ℝ denote the set of all real numbers. Let $z_1 = 1 + 2i$ and $z_2 = 3i$ be two complex numbers, where $i = \sqrt{-1}$. Let

$$S = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + iy - z_1| = 2|x + iy - z_2| \}.$$

Then which of the following statements is (are) TRUE?

Let the set of all relations $R$ on the set $\{a, b, c, d, e, f\}$, such that $R$ is reflexive and symmetric, and $R$ contains exactly $10$ elements, be denoted by $\mathcal{S}$.

Then the number of elements in $\mathcal{S}$ is ________________.

For any two points $M$ and $N$ in the $XY$-plane, let $\overrightarrow{MN}$ denote the vector from $M$ to $N$, and $\vec{0}$ denote the zero vector. Let $P, Q$ and $R$ be three distinct points in the $XY$-plane. Let $S$ be a point inside the triangle $\triangle PQR$ such that

$$\overrightarrow{SP} + 5\; \overrightarrow{SQ} + 6\; \overrightarrow{SR} = \vec{0}.$$

Let $E$ and $F$ be the mid-points of the sides $PR$ and $QR$, respectively. Then the value of

$\frac{\text { length of the line segment } E F}{\text { length of the line segment } E S}$

is ________________.