
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
The Unicom company offers two types of mobile phone plans: Plan $A$ (monthly fee of $15$ yuan, call rate of $0.1$ yuan per minute) and Plan $B$ (monthly fee of $0$ yuan, call rate of $0.15$ yuan per minute). Let $y_{1}$ represent the monthly bill for Plan $A$ (in yuan) and $y_{2}$ represent the monthly bill for Plan $B$ (in yuan), with $x$ representing the monthly call duration in minutes. <br/>$(1)$ Write down the functional relationships between $y_{1}$ and $x$, and $y_{2}$ and $x$; <br/>$(2)$ For how many minutes of call duration in a month will the charges be the same for both Plan $A$ and Plan $B?

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