
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
Billy Bones has two coins - a gold one and a silver one. One of them is symmetric, and the other is not. It is not known which coin is not symmetric, but it is given that the non-symmetric coin lands heads with a probability of $p = 0.6$.

Billy Bones flipped the gold coin, and it landed heads immediately. Then Billy Bones started flipping the silver coin, and heads came up only on the second flip. Find the probability that the gold coin is the non-symmetric one.

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