
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 store sells a certain brand of eye-protection lamp. The cost price of each lamp is $40$ yuan. It is found during the sales process that the monthly sales volume $y$ (units) and the selling price $x$ (yuan) satisfy a linear function relationship. It is stipulated that the selling price is not lower than the cost price and not higher than twice the cost price. Some corresponding data is shown in the table below:

| Selling Price $x$ (yuan) | $\ldots $ | $50$ | $60$ | $70$ | $\ldots $ |
|--------------------------|-----------|------|------|------|-----------|
| Monthly Sales Volume $y$ (units) | $\ldots $ | $90$ | $80$ | $70$ | $\ldots $ |

$(1)$ Find the functional relationship between $y$ and $x$;  
$(2)$ When the selling price of the eye-protection lamp is set to how many yuan, the store will maximize the profit from selling this type of eye-protection lamp each month? What is the maximum monthly profit in yuan?

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