
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 a three-headed dragon named Zmey Gorynych and a battle begins. Each minute, Ilya chops off one of the dragon's heads. With a probability of $\frac{1}{4}$, two new heads grow in place of the chopped-off head; with a probability of $\frac{1}{3}$, only one new head grows; and with a probability of $\frac{5}{12}$, no new heads grow. The dragon is considered defeated if it has no heads left. Find the probability that Ilya will eventually defeat the dragon.

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