
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
A certain factory received a task to install 500 air conditioners. Due to the high time requirement, the factory did not have enough skilled workers and decided to recruit a group of new workers. After the production started, it was found that 1 skilled worker and 3 new workers could install a total of 11 air conditioners per day; 2 skilled workers could install the same number of air conditioners per day as 5 new workers.

$(1)$ Find out how many air conditioners can be installed in one day by 1 skilled worker and 1 new worker together.

$(2)$ If the company originally had m skilled workers and now recruits n new workers (both m and n are not 0), what are the possible solutions to complete the installation task in exactly 20 days?

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