
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 \( y^{2} = 2px \) and a point \( A \) in its plane. From the vertex \( O \) of the parabola we draw an arbitrary chord \( OB \); let \( C \) be the projection of the point \( B \) on the tangent at the vertex. What is the locus of the intersection point \( M \) of the lines \( OB \) and \( AC \)? Determine the nature of this locus depending on whether \( A \) lies on the parabola, within the parabola, or outside the parabola.

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