
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
Endurance of the Dragon. Ilya Muromets encounters a three-headed dragon. Each minute, Ilya chops off one of the dragon's heads. Let \( x \) be the resilience of the dragon (\( x > 0 \)). The probability that \( s \) new heads (\( s = 0, 1, 2 \)) grow back in place of the chopped head is given by

\[ p_{s} = \frac{x^{s}}{1 + x + x^{2}} \]

During the first 10 minutes of the fight, Ilya recorded how many heads regrew in place of each chopped-off head. The resulting vector is:

\[ K = (1, 2, 2, 1, 0, 2, 1, 0, 1, 2) \]

Find the value of the dragon's resilience \( x \) that maximizes the probability of obtaining the vector \( K \).

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