
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 up to 100 alarm system devices per month, with the total revenue for producing $x$ devices ($x \in \mathbb{N}^*$) being $30x - 0.2x^2$ (in ten thousand yuan). The fixed monthly costs (including machinery maintenance, worker salaries, etc.) are 40 ten thousand yuan, and in addition, each device produced incurs a material cost of 5 ten thousand yuan. In economics, the marginal profit function $MP(x)$ of the monthly profit function $P(x)$ is 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 $MP(x)$, determine how many alarm system devices the company should produce each month to achieve maximum profit, and what is the maximum profit?

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