
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 ball is thrown upward from a height \( h \) with an initial velocity \( V_0 \) and accelerates downward due to gravity \( g \). It reaches a final velocity \( V \) after time \( t \). The displacement \( S \) from the initial position after time \( t \) is given by \( S = h + \frac{1}{2}gt^2 + V_0t \). If the final velocity \( V \) is related to \( t \) by \( V = gt + V_0 \), what is the expression for \( t \) in terms of \( S \), \( V \), \( V_0 \), and \( h \)?
A) \( \frac{2S}{V + V_0} \)
B) \( \frac{2(S - h)}{V + V_0} \)
C) \( \frac{2(S + h)}{V - V_0} \)
D) \( \frac{2(S - h)}{V - V_0} \)
E) \( 2S - h \)

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