**Solution:**

Let's break down the problem into three parts:

1. **Initial motion with uniform acceleration**
2. **Uniform motion**
3. **Final motion with uniform retardation**

**1. Initial motion with uniform acceleration:**

Let's assume the initial velocity of the body is u and the acceleration is a.

From the first equation of motion, we know that:

v^2 = u^2 + 2as

Since the body starts from rest, the initial velocity u is zero. Therefore, the equation simplifies to:

v^2 = 2as

We can rearrange the equation to find the value of acceleration:

a = v^2 / (2s)

**2. Uniform motion:**

In this part, the body moves with a uniform velocity for a distance of 2S. Since the velocity is constant, the acceleration is zero.

**3. Final motion with uniform retardation:**

In this part, the body comes to rest after moving a further distance of 5S under uniform retardation. Let's assume the final velocity is v' and the retardation is -a' (negative sign indicates retardation).

Again, using the first equation of motion, we have:

v'^2 = v^2 + 2a's

Since the body comes to rest, the final velocity v' is zero. Therefore, the equation simplifies to:

0 = v^2 + 2a's

We can rearrange the equation to find the value of retardation:

a' = -v^2 / (2s)

**Ratio of average velocity to maximum velocity:**

The average velocity is defined as the total displacement divided by the total time taken.

Total displacement = A + 2S + 5S = 8S

Total time taken = t_1 + t_2 + t_3

The time taken for the initial motion can be calculated using the second equation of motion:

s = ut + (1/2)at^2

Since the initial velocity u is zero, the equation simplifies to:

s = (1/2)at^2

Solving for t:

t_1 = sqrt(2s/a)

The time taken for the final motion can be calculated using the equation:

s = v't - (1/2)a't^2

Since the final velocity v' is zero, the equation simplifies to:

s = -(1/2)a't^2

Solving for t:

t_3 = sqrt(2s/a')

The time taken for uniform motion is simply distance divided by velocity:

t_2 = 2S / v

The total time taken is:

t = t_1 + t_2 + t_3

Now, we can calculate the average velocity:

Average velocity = (Total displacement) / (Total time taken)

Substituting the values:

Average velocity = 8S / (t_1 + t_2 + t_3)

To find the maximum velocity, we need to find the velocity at the end of the initial motion and the velocity at the beginning of the final motion.

Using the first equation of motion for the initial motion:

v = u + at

Since the initial velocity u is zero, the equation simplifies to:

v = at

Using the first equation of motion for the final motion:

v' = v - a't

Since the final velocity v