图示为 $y = a \sin \frac{x}{b} + c$ 的部分图像。 该图像在 $(6\pi, 5)$ 处有一个极大值点，在 $(2\pi, 1)$ 处有一个极小值点。 求常数 $a$、$b$ 和 $c$ 的值。

1. Find the value of $c$ using the maximum and minimum values.

The value of $c$ is the vertical shift, which is the average of the maximum and minimum $y$-values:
$c = \frac{5 + 1}{2} = 3$

2. Find the period of the graph and use it to determine $b$.

The horizontal distance between a minimum and the consecutive maximum is half a period.
$\frac{1}{2} \text{ Period} = 6\pi - 2\pi = 4\pi$
$\text{Period} = 8\pi$
The period of $y = a \sin \frac{x}{b} + c$ is given by $2\pi b$ (assuming $b > 0$).
$2\pi b = 8\pi$
$b = 4$

3. Find the value of $a$ using the amplitude and a point on the graph.

The amplitude is half the difference between the maximum and minimum $y$-values:
$|a| = \frac{5 - 1}{2} = 2$
Substitute the minimum point $(2\pi, 1)$ into the equation $y = a \sin \frac{x}{4} + 3$:
$1 = a \sin \left(\frac{2\pi}{4}\right) + 3$
$1 = a \sin \left(\frac{\pi}{2}\right) + 3$
$1 = a(1) + 3$
$a = -2$