
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
Broady The Boar is playing a boring board game consisting of a circle with  $2021$  points on it, labeled  $0$ ,  $1$ ,  $2$ , ...  $2020$  in that order clockwise. Broady is rolling  $2020$ -sided die which randomly produces a whole number between  $1$  and  $2020$ , inclusive.
    
    Broady starts at the point labelled  $0$ . After each dice roll, Broady moves up the same number of points as the number rolled (point  $2020$  is followed by point  $0$ ). For example, if they are at  $0$  and roll a  $5$ , they end up at  $5$ . If they are at  $2019$  and roll a  $3$ , they end up at  $1$ .
    
    Broady continues rolling until they return to the point labelled  $0$ . What is the expected number of times they roll the dice?


*2021 CCA Math Bonanza Lightning Round #2.3*

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