
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 nearby fruit supermarket near a certain middle school recently received a batch of passion fruits. The purchase price is $8$ yuan per kilogram. In order to set a reasonable price, a flexible pricing strategy was implemented in the first week. The standard selling price was set at $10$ yuan per kilogram. The portion exceeding $10$ yuan was recorded as positive, and the portion below $10$ yuan was recorded as negative. The supermarket recorded the selling price and quantity of passion fruits in the first week as follows:

| Day of the Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Price relative to standard price (yuan per kilogram) | $+1$ | $-2$ | $+3$ | $-1$ | $+2$ | $+5$ | $-4$ |
| Quantity sold (kilograms) | $20$ | $35$ | $10$ | $30$ | $15$ | $5$ | $50$ |

$(1)$ What is the profit or loss of the supermarket from selling this type of passion fruit this week?

$(2)$ In order to promote this type of passion fruit, the supermarket decided to introduce two promotional methods starting from next week:

Method one: Buy up to $5$ kilograms of passion fruit for $12$ yuan per kilogram, with a 20% discount for each kilogram exceeding $5$ kilograms.

Method two: Selling price of $10$ yuan per kilogram.

① If a customer buys $a$ (where $a > 5$) kilograms of passion fruit, express the amount of money needed to buy according to method one and method two using an algebraic expression containing $a$.

② If a teacher decides to buy $35$ kilograms of passion fruit, which method is more cost-effective between the two purchasing methods? Please calculate and explain.

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