
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 Hangzhou Asian Games will be held on September 23, 2023. A sports brand sponsor has developed a new type of sports equipment and plans to produce and install 360 units in 15 days to be delivered to the designated venue for athletes to use. Due to the inability to allocate enough skilled workers to complete the installation of the new sports equipment, the factory decides to recruit some new workers. After training, they can independently install the new sports equipment. After production begins, the research department found that 2 skilled workers and 1 new worker can install 10 units of the new sports equipment per day; 3 skilled workers and 2 new workers can install 16 units of the new sports equipment per day.

$(1)$ How many units of the new sports equipment can each skilled worker and new worker install per day?

$(2)$ If the factory recruits $n (1 < n < 8)$ skilled workers, so that the newly recruited workers (at least one) and the skilled workers can just complete the original 15-day production task, how many recruitment plans are there for the new workers?

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