
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 of a function $f(x)$ is denoted as $Mf(x)$, defined as $Mf(x)=f(x+1)-f(x)$. A company can produce at most $100$ alarm system devices per month. The revenue function for producing $x$ devices is $R(x)=3000x-20x^{2}$ (in yuan), and its cost function is $C(x)=500x+4000$ (in yuan). Profit equals the difference between revenue and cost.
1. Find the profit function $p(x)$ and its marginal profit function $Mp(x)$;
2. Determine if the profit function $p(x)$ and its marginal profit function $Mp(x)$ have the same maximum value;
3. Discuss the practical significance of the maximum value of the marginal profit function $Mp(x)$ in this problem.

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