
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 parabola C: y²=2px (p>0) passes through point M(1, -2).
(I) Find the equation of the parabola C and its axis equation.
(II) The line l passing through point K(-1, 0) intersects C at points A and B. Point A is symmetric to the x-axis at point D. As line l moves around point K, does line BD pass through a fixed point? If so, find it. If not, explain why.

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