
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 equation $16x^{2}-9y^{2}=144$.  
$(1)$ Find the length of the real axis, the length of the imaginary axis, and the eccentricity of the hyperbola;  
$(2)$ If the vertex of the parabola $C$ is the center of the hyperbola, and its focus is the left vertex of the hyperbola, find the equation of the parabola $C$.

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