
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 line with a slope of 1 intersects the x-axis at point \( C \). This line also intersects a parabola, which is symmetric about the x-axis, opens to the right, and passes through the point \((-1,1)\), at points \( A \) and \( B \). The distance \( |AB| = 5\sqrt{2} \), and the distance between point \( C \) and the vertex of the parabola is \( \frac{4}{3} \). Find the equation of the parabola.

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