
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 clothing store deals with two types of T-shirts, $A$ and $B$, with the following purchase prices and selling prices:<br/>

| Item | $A$ | $B$ |
|------|----|----|
| Purchase Price (元/each) | $45$ | $60$ |
| Selling Price (元/each) | $66$ | $90$ |

$(1)$ When making the first purchase, the clothing store spent $6000$ yuan to buy a total of $120$ pieces of T-shirts of types $A$ and $B$. How much profit did they make after selling all the T-shirts?<br/>

$(2)$ Due to market factors, during the second purchase, the purchase price of type $A$ T-shirts increased by $5$ yuan per piece, and the purchase price of type $B$ T-shirts increased by $10$ yuan per piece. However, the selling prices of both types of T-shirts remained the same. The clothing store plans to purchase a total of $150$ pieces of T-shirts of types $A$ and $B$, with the quantity of type $B$ T-shirts not exceeding twice the quantity of type $A$ T-shirts. Let $m$ be the quantity of type $A$ T-shirts purchased this time. If all the T-shirts are sold out, the profit earned will be $W$ yuan.<br/>

① Find the functional relationship between $W$ and $m$;<br/>
② Can the profit from the second purchase at the clothing store exceed the profit from the first purchase? Please explain the reason.

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