
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 shopping mall purchased a total of 50 new energy-saving desk lamps of types $A$ and $B$ for a total of $2500$ yuan. The cost price and the marked price of these two types of desk lamps are shown in the table below:<br/>$(1)$ How many desk lamps of each type were purchased?<br/>$(2)$ If type $A$ desk lamps are sold at a discount of 10% off the marked price, and type $B$ desk lamps are sold at a discount of 30% off the marked price, how much profit does the shopping mall make after selling all these desk lamps?<br/><table class="edittable"><tbody><tr><td width="97" align="center">Price$\backslash $Type</td><td width="63" align="center">Type $A$</td><td width="70" align="center">Type $B$</td></tr><tr><td align="center">Cost Price (yuan per lamp)</td><td align="center">$40$</td><td align="center">$65$</td></tr><tr><td align="center">Marked Price (yuan per lamp)</td><td align="center">$60$</td><td align="center">$100$</td></tr></tbody></table>

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