
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 \( y_1, y_2, \ldots, y_n \) be a sequence of integers such that:
(i) \( -1 \leq y_i \leq 2 \) for \( i = 1, 2, \ldots, n \)
(ii) \( y_1 + y_2 + \cdots + y_n = 23 \)
(iii) \( y_1^2 + y_2^2 + \cdots + y_n^2 = 105 \).
Determine the minimal and maximal possible values of \( y_1^3 + y_2^3 + \cdots + y_n^3 \), and calculate the ratio of the maximum to the minimum value.
A) \(10.7\)
B) \(11.7\)
C) \(12.7\)
D) \(13.7\)

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