
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
An ant moves on the following hexagonal lattice, beginning at the dot labeled $A$. Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random. What is the probability that after 6 minutes he is at the dot labeled $B$? The lattice is arranged such that each dot has up to 6 neighbors in a hexagonal pattern. [asy]
draw((0,0) -- (1,0) -- (0.5,sqrt(3)/2) -- (-0.5,sqrt(3)/2) -- (-1,0) -- (-0.5,-sqrt(3)/2) -- (0.5,-sqrt(3)/2) -- cycle); 
draw((1,0) -- (1.5,sqrt(3)/2));
draw((-1,0) -- (-1.5,-sqrt(3)/2));
dot((0,0)); dot((1,0)); dot((0.5,sqrt(3)/2)); dot((-0.5,sqrt(3)/2)); dot((-1,0)); dot((-0.5,-sqrt(3)/2)); dot((0.5,-sqrt(3)/2)); 
dot((1.5,sqrt(3)/2)); dot((-1.5,-sqrt(3)/2));
label("$A$",(0,0),S);
label("$B$",(-1.5,-sqrt(3)/2),SW);
[/asy]

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