
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 probability \( p \) or an incorrect answer with probability \( q = 1 - p \).

If both advisors give the same answer, the king follows their advice. If they give opposite answers, the king makes a decision by flipping a coin.

One day Arthur thought - why does he need two advisors; wouldn't one be enough? Then the king called the advisors and said:

- It seems to me that the probability of making correct decisions will not decrease if I keep only one advisor and follow his advice. If this is the case, I must fire one of you. If not, I will keep both. Answer me, should I fire one of you?
- Who exactly do you intend to fire, King Arthur? - asked the advisors.
- If I decide to fire one of you, I will make the choice by drawing lots, flipping a coin.

The advisors left to think about the answer. The advisors, we repeat, are equally wise but not equally honest.

Percival is very honest and will try to give the correct answer, even if it threatens his dismissal. Merlin, honest in all other matters, in this situation decides to give an answer that maximizes his 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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