
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 shipbuilding company has a maximum annual shipbuilding capacity of 20 ships. It is known that the revenue function for building $x$ ships is $R(x) = 3700x + 45x^2 - 10x^3$ (in ten thousand yuan), and the cost function is $C(x) = 460x + 500$ (in ten thousand yuan).
(1) Find the profit function $p(x)$; (Hint: profit = revenue - cost)
(2) How many ships should be built annually to maximize the company's annual profit?
(3) In economics, the marginal function of $f(x)$ is defined as $Mf(x) = f(x+1) - f(x)$. Find the marginal profit function $Mp(x)$, and determine the range of $x$ for which $Mp(x)$ is monotonically decreasing; what is the practical significance of $Mp(x)$ being monotonically decreasing in this problem? (Reference formula: $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$)

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