
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
There are 2022 blue balls and 2022 red balls in a box. Ten people numbered from 1 to 10 take turns drawing balls until a red ball is drawn, at which point the next person takes over. If the probability that the person numbered n draws a ball n times is P(n), then find P(1) + P(2) + ∙∙∙ + P(10).

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