
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
With the rapid development of the Internet, the scale of online education users in various countries is increasing. Seizing the opportunity, online teacher Xiao Li started to build a team to produce micro-lesson videos for two different groups of students, $A$ and $B$. It is known that producing 2 $A$ type micro-lessons and 3 $B$ type micro-lessons requires a cost of 2900 yuan, while producing 3 $A$ type micro-lessons and 4 $B$ type micro-lessons requires a cost of 4100 yuan. Teacher Li sells the completed micro-lessons to a video streaming website, with each $A$ type micro-lesson priced at 1500 yuan and each $B$ type micro-lesson priced at 1000 yuan. The team can produce 1 $A$ type micro-lesson or 1.5 $B$ type micro-lessons per day. Additionally, the number of $B$ type micro-lessons produced by the team each month is at least twice the number of $A$ type micro-lessons (Note: the number of $A$ and $B$ type micro-lessons produced each month is an integer). Assuming the team produces micro-lessons for 22 days each month, with $a$ days dedicated to producing $A$ type micro-lessons, the monthly profit from producing both $A$ and $B$ type micro-lessons is $w$ yuan.<br/>$(1)$ Find the cost of producing one $A$ type micro-lesson and one $B$ type micro-lesson respectively.<br/>$(2)$ Find the functional relationship between $w$ and $a$, write the range of values for $a$, and determine the number of $A$ type micro-lessons produced each month that maximizes the team's monthly profit $w$, and what is the maximum profit?

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