
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
The image below consists of a large triangle divided into  $13$  smaller triangles. Let  $N$  be the number of ways to color each smaller triangle one of red, green, and blue such that if  $T_1$  and  $T_2$  are smaller triangles whose perimeters intersect at more than one point,  $T_1$  and  $T_2$  have two different colors. Compute the number of positive integer divisors of  $N$ .

[asy]
size(5 cm);
        draw((-4,0)--(4,0)--(0,6.928)--cycle);
        draw((0,0)--(2,3.464)--(-2,3.464)--cycle);
        draw((-2,0)--(-1,1.732)--(-3,1.732)--cycle);
        draw((2,0)--(1,1.732)--(3,1.732)--cycle);
        draw((0,3.464)--(1,5.196)--(-1,5.196)--cycle);
[/asy]


*2021 CCA Math Bonanza Individual Round #7*

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