
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
At a certain commodity trading fair, a merchant sells a souvenir with a cost price of $5$ yuan for $9$ yuan each, and can sell $32$ items per day. The merchant wants to increase profits by raising the selling price. Through experimentation, it was found that for every $1$ yuan increase in the selling price of the souvenir, the daily sales volume decreases by $4$ items.<br/>$(1)$ If the selling price is increased by $x$ yuan $(x$ is a positive integer), write the functional relationship between the daily sales volume $y$ (in items) and the price increase $x$ (in yuan).<br/>$(2)$ If the daily profit from selling this type of souvenir is $w$ yuan, in order to maximize the profit obtained per day, at what price should the souvenir be sold per unit? What is the maximum profit per day?

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