
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
Let us switch to a reference frame that is falling with an acceleration \( g \) and has an initial velocity of zero. Time starts at the moment the apple is thrown. In this chosen reference frame, both the apple and the arrow move uniformly in straight lines according to the principle of acceleration composition. Therefore, the point at which the arrow will hit the apple is the third vertex \( C \) of a triangle with the base \( AB \) and angles \( \alpha \) and \( \beta \). The length \( AC \) can be found using the law of sines:
\[ AC = L \frac{\sin \beta}{\sin (\alpha + \beta)} \]
The sought time \( T \) is given by:
\[ T = \frac{AC}{V_0} = \frac{L}{V_0} \frac{\sin \beta}{\sin (\alpha + \beta)} = \frac{3}{4} \text{ seconds} \]

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