
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
We throw a heavy object vertically upward from point $A$ with an initial velocity of \( v_{0} \). At the same time, from point $B$, which is $a$ meters above $A$, we drop another heavy object without an initial velocity. After how many seconds will the two objects meet? How much distance has each object traveled? Determine the conditions under which the two objects meet:

1.) Between points $A$ and $B$,
2.) At point $A$,
3.) Below point $A$.

Example values: \( v_{0} = 30 \mathrm{~m/s}, \quad a = 420 \mathrm{~m}, \quad g = 9.8 \mathrm{~m/s^2} \).

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