
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 new PUMaC tournament hosts  $2020$  students, numbered by the following set of labels  $1, 2, . . . , 2020$ . The students are initially divided up into  $20$  groups of  $101$ , with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the second round. Out of these  $20$  students, we chose the champion uniformly at random. If the expected value of champion’s number can be written as  $\frac{a}{b}$ , where  $a, b$  are relatively prime integers, determine  $a + b$ .

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