
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
A triangle has a fixed base $AB$ that is $6$ cm long. The median from vertex $A$ to side $BC$ measures $4$ cm and altitude from $A$ to $BC$ is also given as $3$ cm. The locus of vertex $C$ is:
A) A circle with radius 3 cm and center 3 cm from B
B) A circle with radius 3 cm and center 4 cm from A along a line perpendicular to AB
C) An ellipse with A as the focus
D) A circle with A as the center and radius 9 cm

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