
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 a parabola $C$ whose focus is the left vertex of the hyperbola $16x^2 - 9y^2 = 144$:
1. Determine the standard equation of the parabola $C$.
2. If a line passing through the point $(0,2)$ with a slope of $1$ intersects the parabola $C$ at points $A$ and $B$, find the length of the segment $AB$.

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