
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 \\({a}\_{1},{a}\_{2},\ldots,{a}\_{50} \\) be a sequence of numbers chosen from the integers \\(-1\\), \\(0\\), and \\(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 \\), then the number of \\(0\\)s in \\({a}\_{1},{a}\_{2},\ldots,{a}\_{50} \\) is \_\_\_\_\_\_.

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