
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 company can produce a maximum of $100$ alarm system devices per month. The total revenue from producing $x$ devices $(x∈N^{*})$ is given by $30x-0.2x^{2}$ (in ten thousand units). The fixed monthly cost, which includes machine maintenance and employee salaries, is $40$ ten thousand units. Additionally, each device produced requires a material cost of $5$ ten thousand units. In economics, the profit function $P(x)$ and its marginal profit function $MP(x)$ are often used to study when the maximum profit is obtained, where $MP(x)=P(x+1)-P(x)$.

(I) Find the profit function $P(x)$ and its marginal profit function $MP(x)$;

(II) Using the marginal profit function $MP(x)$, investigate how many alarm system devices should the company produce per month to obtain the maximum profit? What is the maximum profit?

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