
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
China Unicom charges for mobile phone calls with two types of packages: Package $A$ (monthly fee of $15$ yuan, call fee of $0.1 yuan per minute) and Package $B$ (monthly fee of $0$ yuan, call fee of $0.15 yuan per minute). Let $y_{1}$ represent the monthly bill for Package $A$ (in yuan), $y_{2}$ represent the monthly bill for Package $B$ (in yuan), and $x$ represent the monthly call duration in minutes. <br/>$(1)$ Express the functions of $y_{1}$ with respect to $x$ and $y_{2}$ with respect to $x$. <br/>$(2)$ For how long should the monthly call duration be such that the charges for Packages $A$ and $B$ are the same? <br/>$(3)$ In what scenario is Package $A$ more cost-effective?

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