
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 part, with each part having a cost of $40$ and an ex-factory price of $70$. To encourage sales agents to order more, the factory decides that when the order quantity exceeds $100$, the ex-factory price of all parts will decrease by $0.02$ yuan for each additional order. However, the actual ex-factory price cannot be lower than $61$ yuan.
1. Express the actual ex-factory price $y$ in terms of the order quantity $x$ as a function $y=f(x)$.
2. What is the factory's profit when a sales agent orders $500$ parts at once?

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