
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
To commend students who have performed excellently in the themed event "Commemorating the May Fourth Movement," a certain class needs to purchase $10$ backpacks and at least $10$ stationery boxes. A stationery store has formulated two discount schemes: ① Buy one backpack and get one stationery box for free, with additional stationery boxes beyond the number of backpacks charged at the original price; ② Both backpacks and stationery boxes are charged at 10% off the original price. It is known that each backpack is priced at $40, and each stationery box is priced at $5. <br/> $(1)$ Let the number of stationery boxes to be purchased be $x$. The cost of purchasing under scheme ① is $y_{1}$ yuan, and under scheme ② is $y_{2}$ yuan. Please write down the relationship between $y_{1}$, $y_{2}$, and $x$ respectively; <br/> $(2)$ How many stationery boxes need to be purchased for the costs under both schemes to be the same?

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