
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 hyperbola $C_1$ with eccentricity $e_1$ has real semi-axis $a$ and imaginary semi-axis $b (a > b)$. Both semi-axes are simultaneously increased by $m (m > 0)$ units, resulting in a hyperbola $C_2$ with eccentricity $e_2$. Determine the relationship between $e_1$ and $e_2$.

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