
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 factory produces a certain type of component, with the production cost per unit being 40 yuan, and the set factory price being 60 yuan. To encourage orders from merchants, the factory decided that for each order exceeding 100 units, the factory price per unit would decrease by 0.02 yuan for each additional unit ordered; however, the actual factory price per unit cannot fall below 51 yuan.

(1) How many units should be ordered in one purchase for the actual factory price per unit to be exactly 51 yuan?
(2) Let the order quantity be $x$ units, and the actual factory price be $P$ yuan. Write the expression for the function $P=f(x)$.
(3) How much profit does the factory make when a merchant orders 500 units at once? And what would the profit be for an order of 1,000 units? (The profit for the factory from selling one unit is the actual factory price minus the cost)

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