
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. An interior point  $P$  of  $ABC$  is said to be *good*if we can find exactly  $27$  rays emanating from  $P$  intersecting the sides of the triangle  $ABC$  such that the triangle is divided by these rays into  $27$  *smaller triangles of equal area.* Determine the number of good points for a given triangle  $ABC$ .

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