
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 advisor finds the correct answer to any question with probability \( p \) and the wrong 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 wondered if he needed two advisors or if one would suffice. He then summoned the advisors and said:

- It seems to me that the probability of making the correct decisions won't decrease if I keep only one advisor and follow their advice. If this is true, I must dismiss one of you. If not, I will keep everything as it is. Answer me, should I dismiss one of you?

- Which one do you intend to dismiss, King Arthur? - asked the advisors.
- If I decide to dismiss one of you, I will make the choice by flipping a coin.

The advisors went away to think about their answer. The advisors, we repeat, are equally wise but not equally honest. Percival is very honest and will try to give the true answer, even if it means he faces dismissal. Merlin, honest in all other matters, decides to give such an answer that minimizes the probability of his dismissal. What is the probability that Merlin will be dismissed?

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