
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
Let  $ A$  and  $ B$  be fixed distinct points on the  $ X$  axis, none of which coincides with the origin  $ O(0, 0),$  and let  $ C$  be a point on the  $ Y$  axis of an orthogonal Cartesian coordinate system. Let  $ g$  be a line through the origin  $ O(0, 0)$  and perpendicular to the line  $ AC.$  Find the locus of the point of intersection of the lines  $ g$  and  $ BC$  if  $ C$  varies along the  $ Y$  axis. Give an equation and a description of the locus.

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