
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
Triangle  $\vartriangle ABC$  has circumcenter  $O$  and orthocenter  $H$ . Let  $D$  be the foot of the altitude from  $A$  to   $BC$ , and suppose  $AD = 12$ . If  $BD = \frac14 BC$  and   $OH  \parallel  BC$ , compute  $AB^2$ .
.

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