
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
In $\triangle ABC$, $\cos \frac{C}{2} = \frac{2\sqrt{5}}{5}$, $\overrightarrow{A H} \cdot \overrightarrow{BC} = 0$, and $\overrightarrow{AB} \cdot (\overrightarrow{CA} + \overrightarrow{CB}) = 0$.

Find the eccentricity of the hyperbola passing through point $C$ and having $A$ and $H$ as its foci.

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