
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 zoo launched two commemorative badges, $A$ and $B$, on the occasion of its anniversary celebration. It is known that the cost price of each $A$ badge is $4$ yuan more than the cost price of each $B$ badge. The cost of purchasing $6$ $A$ badges is the same as the cost of purchasing $10$ $B$ badges. The selling price of $A$ badges is $13$ yuan each, and the selling price of $B$ badges is $8$ yuan each.  
$(1)$ What is the cost price of each $A$ badge and each $B$ badge?  
$(2)$ According to online reservations, the zoo plans to purchase a total of $400$ $A$ and $B$ badges with a budget of no more than $2800$ yuan. All these $400$ badges can be sold. Which purchasing plan will maximize the zoo's profit? What is the maximum profit?

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