
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
A pirate now explores 8 islands in search of treasure. Each island independently has a $\frac{1}{3}$ chance of containing treasure and no traps, a $\frac{1}{6}$ chance of containing traps but no treasure, and a $\frac{1}{2}$ chance of having neither traps nor treasure. Determine the probability that the pirate finds exactly 4 islands with treasure, and none of these islands contain traps, while searching all 8 islands.

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