
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
You have a large number of congruent equilateral triangular tiles on a table and you want to fit \( n \) of them together to make a convex equiangular hexagon (i.e., one whose interior angles are \( 120^{\circ} \) ). Obviously, \( n \) cannot be any positive integer. The first three feasible \( n \) are 6, 10, and 13. Determine if 19 and 20 are feasible.

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