
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
A token starts at the point  $(0,0)$  of an  $xy$ -coordinate grid and them makes a sequence of six moves. Each move is  $1$  unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a point on the graph of  $|y|=|x|$  is  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

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