
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 experimental field plans to plant a new type of crop. After investigation, it was found that the total cost $y$ (in million yuan) of planting $x$ acres consists of three parts: machinery cost, management cost, and other costs. The machinery cost is fixed at 10 million yuan, the management cost (in million yuan) is directly proportional to $x$, and the other cost (in million yuan) is proportional to the square of $x$. The data obtained during the production process is as shown in the table below:

| $x$ (acres) | 1 | 3 |
|-------------|---|---|
| $y$ (million yuan) | 16 | 34 |

$(1)$ Find the functional relationship between $y$ and $x$;  
$(2)$ Given that the average cost per acre is 12 million yuan, determine how many acres of the new crop should be planted;  
$(3)$ If the revenue per acre is 15 million yuan, at what value of $x$ will the total profit of the experimental field be maximized, and what is the maximum profit? (Note: Total profit = Total revenue - Total cost)

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