
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 fruit company transports a batch of fruits to a certain location. The transportation cost by company A is $0.6$ yuan per kilogram, while the cost by company B is half of company A's cost per kilogram, but an additional cost of $750$ yuan is required after transporting this batch of fruits. Let the company transport $x$ kilograms of fruits $(0 < x < 5000)$. The cost of choosing company A for transportation is $y_{1}$ yuan, and the cost of choosing company B for transportation is $y_{2}$ yuan.  
$(1)$ Write down the functional relationships between $y_{1}$, $y_{2}$, and $x$.  
$(2)$ Which transportation company should the fruit company choose for lower cost? Please explain the reason.

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