
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
A merchant usually determines the selling price of a product based on the "optimistic coefficient rule," which involves the minimum selling limit $a$, the maximum selling limit $b (b > a)$, and a constant $x (0 < x < 1)$ to determine the actual selling price $c = a + x(b - a)$. Here, $x$ is called the optimistic coefficient.
Empirical evidence shows that the best optimistic coefficient $x$ makes $(c - a)$ the geometric mean of $(b - c)$ and $(b - a)$. Based on this, find the value of the best optimistic coefficient $x$.

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