
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
Businesses often determine the price of goods for sale based on the "Optimism Coefficient Criterion". They calculate the actual selling price $c$ according to the product's minimum price limit $a$, maximum price limit $b$ (where $b > a$), and a constant $x$ (where $0 < x < 1$), using the formula $c = a + x(b - a)$. Here, $x$ is referred to as the optimism coefficient.

Experience shows that the optimal optimism coefficient $x$ makes $(c - a)$ the geometric mean of $(b - c)$ and $(b - a)$. Based on this, the value of the optimal optimism coefficient $x$ is equal to __________.

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