
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 starts at a dot labeled $A$ on a lattice and moves every minute to a neighboring dot, choosing randomly among its neighbors. Determine the probability that after 4 minutes, the ant is at the dot labeled $C$. The lattice and its coloring are as follows: [asy]
draw((-2,0)--(2,0));
draw((0,-2)--(0,2));
draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle);
dot((0,0)); dot((1,0)); dot((2,0)); dot((-1,0)); dot((-2,0)); dot((0,1)); dot((0,2)); dot((0,-1)); dot((0,-2)); dot((1,1)); dot((1,-1)); dot((-1,-1)); dot((-1,1));
label("$A$",(0,0),SW);
label("$C$",(1,0),NE);
[/asy]
The dots are colored according to the following pattern: dots on even x+y coordinates are red, and dots on odd x+y coordinates are blue.

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