
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
Every spring, the H1N1 flu virus spreads widely, becoming one of the current mainstream epidemic diseases. In a certain community, one resident accidentally contracted the H1N1 flu. Due to the rapid spread of the H1N1 flu, after two rounds of infection, a total of 36 people in the community have contracted the H1N1 flu.
- $(1)$ On average, how many people does one person infect in each round of infection?
- $(2)$ If the infection continues at this rate, will the cumulative number of people with the H1N1 flu exceed 200 after three rounds of infection?

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