
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 furniture factory produces a type of desk, with each desk costing 50 yuan to make and having a factory price set at 80 yuan. To encourage retailers to place larger orders, the factory decides that for orders exceeding 100 desks, the factory price of each desk in that order will be reduced by 0.02 yuan for every desk over 100. According to market research, the quantity ordered by retailers will not exceed 1000 desks.  
(Ⅰ) Let the quantity ordered at one time be $x$ desks, and the actual factory price be $P$ yuan. Find the function $P(x)$ that describes the relationship between $P$ and $x$;  
(Ⅱ) When the quantity ordered $x$ is how much, will the furniture factory's profit $f(x)$ from selling the desks be maximized? What is the maximum profit in yuan? (The profit from selling one desk = actual factory price - cost).

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