
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
Divide a regular hexagon into  $24$  identical small equilateral triangles as shown. To each of the  $19$  points that are vertices of at least one of the equilateral triangles, a different number is assigned. Find the maximum possible number of small equilateral triangles with their vertices having numbers in ascending order going clockwise.

[asy]
size(5.5cm);
int n=6;
pair[] V= sequence(new pair(int i){return dir(360*i/n);}, n);
V.cyclic=true;
for(int i=0;i<n;++i){
  draw(V[i]--V[i+1],black+0.7bp);
  draw(V[i]--V[i+3],black+0.7bp);
  draw(midpoint(V[i]--V[i+1])--midpoint(V[i+2]--V[i+3]),black+0.7bp);
}
[/asy]

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