1. **Determine the coordinates of points $A$, $B$, and $C$**:
- Let $B = (0,0)$.
- Let $C = (x,0)$ for some $x$.
- Let $A = (a,b)$.

2. **Find the coordinates of points $D$, $E$, and $F$**:
- Since $BC:CD = 1:1$, $D$ is at $(2x,0)$.
- Since $CA:AE = 1:2$, $E$ is at $(-\frac{a}{2}, -\frac{b}{2})$.
- Since $AB:BF = 1:3$, $F$ is at $(3a, 3b)$.

3. **Find the centroid $G$ of triangle $ABC$**:
- The centroid $G$ of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$ is given by:
$$G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)$$
- For triangle $ABC$:
$$G = \left( \frac{0 + x + a}{3}, \frac{0 + 0 + b}{3} \right)$$
- Given $G = (32, 24)$, we have:
$$\frac{x + a}{3} = 32 \quad \text{and} \quad \frac{b}{3} = 24$$
$$x + a = 96 \quad \text{and} \quad b = 72$$

4. **Find the centroid $K$ of triangle $DEF$**:
- For triangle $DEF$:
$$K = \left( \frac{2x - \frac{a}{2} + 3a}{3}, \frac{0 - \frac{b}{2} + 3b}{3} \right)$$
$$K = \left( \frac{2x + \frac{5a}{2}}{3}, \frac{\frac{5b}{2}}{3} \right)$$
$$K = \left( \frac{4x + 5a}{6}, \frac{5b}{6} \right)$$

5. **Substitute the values of $a$ and $b$**:
- From step 3, $b = 72$:
$$K = \left( \frac{4x + 5a}{6}, \frac{5 \cdot 72}{6} \right)$$
$$K = \left( \frac{4x + 5a}{6}, 60 \right)$$

6. **Determine the coordinates of $K$**:
- From step 3, $x + a = 96$:
$$K = \left( \frac{4x + 5(96 - x)}{6}, 60 \right)$$
$$K = \left( \frac{4x + 480 - 5x}{6}, 60 \right)$$
$$K = \left( \frac{480 - x}{6}, 60 \right)$$

7. **Find the distance $GK$**:
- $G = (32, 24)$ and $K = \left( \frac{480 - x}{6}, 60 \right)$:
$$GK = \sqrt{\left( 32 - \frac{480 - x}{6} \right)^2 + (24 - 60)^2}$$
$$GK = \sqrt{\left( \frac{192 - 480 + x}{6} \right)^2 + (-36)^2}$$
$$GK = \sqrt{\left( \frac{-288 + x}{6} \right)^2 + 1296}$$
$$GK = \sqrt{\left( \frac{x - 288}{6} \right)^2 + 1296}$$
$$GK = \sqrt{\frac{(x - 288)^2}{36} + 1296}$$
$$GK = \sqrt{\frac{(x - 288)^2 + 46656}{36}}$$
$$GK = \frac{\sqrt{(x - 288)^2 + 46656}}{6}$$

8. Simplify the expression:
- Since $x = 96 - a$ and $a = 96 - x$, we can substitute $x = 96$:
$$GK = \frac{\sqrt{(96 - 288)^2 + 46656}}{6}$$
$$GK = \frac{\sqrt{(-192)^2 + 46656}}{6}$$
$$GK = \frac{\sqrt{36864 + 46656}}{6}$$
$$GK = \frac{\sqrt{83520}}{6}$$
$$GK = \frac{288}{6}$$
$$GK = 48$$

The final answer is $\boxed{48}$.