
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
In April of this year, avian influenza broke out in many countries, leading to a large number of laying hens being culled, causing a global "egg shortage." If a laying hen in a certain country is infected with avian influenza and a total of 64 laying hens are infected after two rounds of infection, find:<br/>$(1)$ How many healthy laying hens on average does each infected laying hen infect in each round of infection?<br/>$(2)$ If not controlled in time, will the number of infected laying hens exceed 500 after three rounds of infection?

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