
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 \( \triangle ABC \), it is known that
\[ \cos C = \frac{2 \sqrt{5}}{5} \]
\[ \overrightarrow{A H} \cdot \overrightarrow{BC} = 0 \]
\[ \overrightarrow{AB} \cdot (\overrightarrow{CA} + \overrightarrow{CB}) = 0 \]

Determine the eccentricity of the hyperbola passing through point \( C \) with foci at points \( A \) and \( H \).

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