
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 with the vertex at the origin and the axis of symmetry along the x-axis, the curve passes through point A $(\frac{1}{2}, -\sqrt{2})$.
(Ⅰ) Find the standard equation of the parabola.
(Ⅱ) Line $l$ passes through the fixed point $P(-2, 1)$ with a slope $k$. What is the value of $k$ when line $l$ intersects the parabola at two distinct points?

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