
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
How many triangles are in the following triangular grid? [asy]
defaultpen(linewidth(0.7));
pair hexcoords (real over, real upover)
{
    return dir(0)*over+dir(60)*upover;
}
real r = 0.3;
int i,j;
for(i=0;i<=3;++i) // Increase the grid size
{
    for(j=0;j<=3-i;++j)
    {
        draw(hexcoords(i,j)--hexcoords(i+1,j));
        draw(hexcoords(i,j)--hexcoords(i,j+1));
        draw(hexcoords(i+1,j)--hexcoords(i,j+1));
    }
}
draw(hexcoords(1,3)--hexcoords(1,4)--hexcoords(-1,4)--hexcoords(0,3)); // Extend the outer triangle
[/asy]

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