
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 purchases a batch of T-shirts at a price of $30$ yuan each. If they are sold at a price of $40$ yuan each, 300 pieces can be sold in a month. Based on past sales experience, for every $1$ yuan increase in the selling price, the sales volume decreases by 10 pieces. Let $x$ represent the increase in the selling price of the T-shirts.<br/>$(1)$ The clothing store hopes to make a profit of $3360$ yuan from selling this type of T-shirt in a month and reduce the inventory as much as possible. How much should the selling price of the T-shirt be increased by?<br/>$(2)$ At what price should the selling price be set to maximize the profit earned by selling this type of T-shirt in a month? What is the maximum profit earned?

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