
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
Arthur stands on a circle drawn with chalk in a parking lot.  It is sunrise and there are birds in the trees nearby.  He stands on one of five triangular nodes that are spaced equally around the circle, wondering if and when the aliens will pick him up and carry him from the node he is standing on.  He flips a fair coin  $12$  times, each time chanting the name of a nearby star system.  Each time he flips a head, he walks around the circle, in the direction he is facing, until he reaches the next node in that direction.  Each time he flips a tail, he reverses direction, then walks around the circle until he reaches the next node in that new direction.  After  $12$  flips, Arthur finds himself on the node at which he started.  He thinks this is fate, but Arthur is quite mistaken.  If  $a$  and  $b$  are relatively prime positive integers such that  $a/b$  is the probability that Arthur flipped exactly  $6$  heads, find  $a+b$ .

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