
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
The game "Geographic Dice" involves each participant rolling a die a number $n$ of times and noting the sequence, with the first roll indicating the distance, in meters, that the participant will walk to the North, the second to the East, the third to the South, the fourth to the West, the fifth to the North, and so on. After this, the distance to the origin is measured. The one who gets closest to the initial position wins.

a) Márcia rolled 214365 (6 rolls). What is her distance from the origin?
b) In 4 rolls, how many possibilities exist for the participant to return to the origin after the last roll?
c) In 5 rolls, how many possibilities exist for the participant to return to the origin after the last roll?

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