
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  $ABC$  be a triangle.  The incircle  $\omega$  of  $\triangle ABC$ , which has radius  $3$ , is tangent to  $\overline{BC}$  at  $D$ .  Suppose the length of the altitude from  $A$  to  $\overline{BC}$  is  $15$  and  $BD^2 + CD^2 = 33$ .  What is  $BC$ ?

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