
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
Ilya Muromets encounters the three-headed Dragon, Gorynych. Each minute, Ilya chops off one head of the dragon. Let $x$ be the dragon's resilience ($x > 0$). The probability $p_{s}$ that $s$ new heads will grow in place of a chopped-off one ($s=0,1,2$) is given by $\frac{x^{s}}{1+x+x^{2}}$. During the first 10 minutes of the battle, Ilya recorded the number of heads that grew back for each chopped-off one. The vector obtained 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 vector $K$.

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