
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
Using $1$ piece of type $A$ steel plate can be exactly made into $2$ pieces of type $C$ steel plate and $l$ pieces of type $D$ steel plate; using $1$ piece of type $B$ steel plate can be exactly made into $1$ piece of type $C$ steel plate and $3$ pieces of type $D$ steel plate.  
$(1)$ If $14$ pieces of type $C$ steel plate and $12$ pieces of type $D$ steel plate are needed, how many pieces of type $A$ and type $B$ steel plates are required?  
$(2)$ Now, $50$ pieces of type $A$ and type $B$ steel plates are prepared and all processed into type $C$ and type $D$ steel plates. It is required that the number of type $C$ steel plates does not exceed $86$ pieces and the number of type $D$ steel plates does not exceed $90$ pieces. How many different purchasing plans are there for type $A$ and type $B$ steel plates?  
$(3)$ Under the conditions of $(2)$, if the profit per piece of type $C$ steel plate is $100$ yuan and the profit per piece of type $D$ steel plate is $120$ yuan, what is the maximum profit that can be obtained by selling all type $C$ and type $D$ steel plates?

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