
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 juggler throws a ball upward every second; with what initial speed must the balls be thrown so that the first ball falls back into his hand when

a) the second ball is thrown,

b) the third ball is thrown?

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