
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A body starts from an initial position \( S_0 \) with an initial velocity \( V_0 \) and moves under the influence of gravity \( g \) and a friction coefficient \( \mu \). The velocity and displacement of the body at time \( t \) are given by:
\[ V = gt + V_0 - \mu g t \]
\[ S = S_0 + \frac{1}{2}gt^2 + V_0t - \frac{1}{2}\mu g t^2 \]
Find the time \( t \) at which the displacement \( S \) equals:
\[ S = S_0 + 100 \]
**A)** \( \frac{-V_0 + \sqrt{V_0^2 + 200(1-\mu)g}}{(1-\mu)g} \)
**B)** \( \frac{-V_0 - \sqrt{V_0^2 + 200(1-\mu)g}}{(1-\mu)g} \)
**C)** \( \frac{V_0 + \sqrt{V_0^2 + 200(1-\mu)g}}{(1-\mu)g} \)
**D)** \( \frac{V_0 - \sqrt{V_0^2 + 200(1-\mu)g}}{(1-\mu)g} \)

Present the answer in LaTex format: \boxed{Your answer}
user