
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

King Arthur has two equally wise advisors – Merlin and Percival. Each of them finds the correct answer to any question with a probability \( p \), or an incorrect answer with a probability \( q = 1-p \).

If both advisors give the same answer, the king listens to them. If they give opposite answers, the king makes a decision by flipping a coin.

One day, Arthur wondered whether he needed two advisors or if one would suffice. He called the advisors and said:

- I think the probability of making the right decisions won't decrease if I only keep one advisor and follow him. If this is correct, I should fire one of you. If not, I will leave everything as it is. Answer me, should I fire one of you?

- Whom exactly are you going to fire, King Arthur? – asked the advisors.
- If I decide to fire one of you, I will choose by drawing lots, flipping a coin.

The advisors left to think about the answer. The advisors, it is to be repeated, are equally wise, but not equally honest.

Percival is very honest and will try to give the correct answer, even if he risks being fired. But Merlin, honest in all other matters, in this situation decides to give an answer that maximizes the probability of not being fired. What is the probability that Merlin will be fired?

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