
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
In economics, the marginal function $Mf(x)$ of a function $f(x)$ is defined as $Mf(x) = f(x+1) - f(x)$. A company can produce up to 100 alarm systems per month, and the revenue function for producing $x$ ($x \in \mathbb{N}^*$) units is $R(x) = 3000x + ax^2$ (unit: Yuan), and its cost function is $C(x) = kx + 4000$ (unit: Yuan), where profit is the difference between revenue and cost. When 10 units are produced, the cost is 9000 Yuan, and the profit is 19000 Yuan.  
(1) Find the profit function $P(x)$ and the marginal profit function $MP(x)$;  
(2) Do the profit function $P(x)$ and the marginal profit function $MP(x)$ have the same maximum value?

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