
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 is searching for buried treasure on 8 islands. Each island has a $\frac{1}{4}$ probability of containing buried treasure and no traps, a $\frac{1}{12}$ probability of having traps but no treasure, and a $\frac{2}{3}$ probability of having neither traps nor treasure. What is the probability that, while searching all 8 islands, the pirate encounters exactly 5 islands with treasure and none with traps?

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