
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 $x_1, x_2, \ldots , x_n$ be a sequence of integers such that $-1 \le x_i \le 2$ for $i = 1,2, \ldots n$, $x_1 + \cdots + x_n = 19$ and $x_1^2 + x_2^2 + \cdots + x_n^2 = 99$, determine the ratio of the maximal possible value of $x_1^3 + \cdots + x_n^3$ to the minimal possible value of $x_1^3 + \cdots + x_n^3$.

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