
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
Given a sequence $a_1, a_2, \ldots, a_{50}$ of integers taken from the set ${-1, 0, 1}$, if $a_1 + a_2 + \ldots + a_{50} = 9$ and $(a_1 + 1)^2 + (a_2 + 1)^2 + \ldots + (a_{50} + 1)^2 = 107$, determine the number of $0$'s in the sequence $a_1, a_2, \ldots, a_{50}$.

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