
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 subsidiary company sells a brand of products, with each product costing $3$ yuan. Additionally, a management fee of $3$ yuan per product must be paid to the head office. It is estimated that when the selling price of each product is $x (9 \leqslant x \leqslant 11)$ yuan, the annual sales volume will be $(12 - x)^2$ ten thousand units.

$(1)$ Find the function $y(\text{ten thousand yuan})$ representing the annual profit of the subsidiary company in relation to the selling price of each product.
$(2)$ Determine the selling price of each product that maximizes the subsidiary company's annual profit $y$, and find the maximum value of $y$.

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