
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
Consider the parabola $y=x^{2}+ax-5$. Take two points on the parabola with x-coordinates $x_1=-4$ and $x_2=2$. Draw a secant line through these two points, and there exists a line parallel to this secant that is tangent to both the parabola and the circle $5x^{2}+5y^{2}=36$. Determine the coordinates of the vertex of the parabola.

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