
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
Given the hyperbola $$C: \frac{x^{2}}{m+1} - \frac{y^{2}}{m} = 1 (m > 0)$$ whose left focus $F$ lies on the circle $x^2 + y^2 - 2x - 6y - 15 = 0$, find the eccentricity of the hyperbola $C$.

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