
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 vertically upward with initial velocity $V_0$. After $t_1$ seconds, it attains a maximum height, $H$, then it starts falling down due to gravity ($g$). At total time $t$, the ball returns to the launch point. If $V = -V_0$ when it returns to the launch point, and $H = \frac{1}{2}gt_1^2$, find $t$.
**A)** $\frac{2V_0}{g}$  
**B)** $\frac{V_0}{2g}$  
**C)** $\frac{V_0^2}{g}$  
**D)** $\frac{2g}{V_0}$

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