
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 and  $n$  a positive integer. Consider on the side  $BC$  the points  $A_1, A_2, ..., A_{2^n-1}$  that divide the side into   $2^n$  equal parts, that is,  $BA_1=A_1A_2=...=A_{2^n-2}A_{2^n-1}=A_{2^n-1}C$ . Set the points  $B_1, B_2, ..., B_{2^n-1}$  and  $C_1, C_2, ..., C_{2^n-1}$  on the sides  $CA$  and  $AB$ , respectively, analogously. Draw the line segments  $AA_1, AA_2, ..., AA_{2^n-1}$ ,  $BB_1, BB_2, ..., BB_{2^n-1}$  and  $CC_1, CC_2, ..., CC_{2^n-1}$ . Find, in terms of  $n$ , the number of regions into which the triangle is divided.


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