
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
During the Hangzhou Asian Games, a certain store sold a batch of Asian Games mascot pendants, each costing $13$ yuan. It was stipulated that the selling price should not be less than $20$ yuan. During the sales period, it was found that when the selling price was set at $20$ yuan, 200 units could be sold per month. For every increase of $1$ yuan in the selling price, the daily sales volume decreased by 10 units. Now the store has decided to raise the price for sales.
- $(1)$ By how much should the price be increased so that the profit is $1620$ yuan?
- $(2)$ At what price should the Asian Games mascot pendants be sold so that the store can maximize the profit obtained from the daily sales of the pendants? What is the maximum profit in this case?

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