
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 certain school needs to purchase a batch of office desks, and companies $A$ and $B$ are both willing to become suppliers of these office desks. After investigation, it is found that the quality and unit price of the office desks produced by the two companies are the same, that is, each office desk costs $500$ yuan. After negotiation, the discount condition offered by company $A$ is that all office desks are discounted by 10% of the unit price, but the school needs to bear a shipping cost of $1000$ yuan; the discount condition offered by company $B$ is that the selling price of each office desk remains the same, but the company bears the shipping cost. Let $x$ represent the number of office desks that the school needs to purchase, $y_{1}$ represent the total cost when purchasing from company $A$, and $y_{2}$ represent the total cost when purchasing from company $B$. <br/>$(1)$ Find the functional relationships between $y_{1}$, $y_{2}$, and $x$. <br/>$(2)$ Which company should the school choose to purchase office desks from? Please explain the reason.

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