
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 a cost of 40 yuan per component and a factory price of 60 yuan. To encourage orders from sales agents, the factory decides that if an order exceeds 100 units, the factory price of all components in the order will decrease by 0.02 yuan for each additional unit ordered, but the actual factory price cannot be lower than 51 yuan.

 (1) When the order quantity is how many units, will the actual factory price of the components just drop to 51 yuan?

 (2) Let the order quantity be \(x\) units, and the actual factory price be \(p\) yuan, write the analytical expression for the function \(p=f(x)\);

 (3) When a sales agent orders 500 components at once, what is the profit for the factory? What about if 1000 components are ordered? (The profit of the factory from selling one component = actual factory price - cost per component)

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