
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
In deep memory of the beloved Premier Li Keqiang, four years ago in the government work report on "further reducing communication costs," he pointed out that the average tariff for mobile network traffic should be reduced by more than $20\%$, implementing "number portability" nationwide, standardizing package settings, truly reducing fees, and ensuring that our people benefit clearly and substantially. A certain communication operator actively responded to the national call and launched two charging methods for mobile calls, $A$ and $B$, as shown in the table below.

| Charging Method | Monthly Call Fee ($/$) | Included Call Time ($/\min$) | Overtime Fee ($/($/min$)$) |
|-----------------|------------------------|-----------------------------|---------------------------|
| $A$             | $30$                   | $600$                       | $0.1$                     |
| $B$             | $50$                   | $1200$                      | $0.1$                     |

$(1)$ Let the monthly call time be $x$ minutes. Then, the charges $y_{1}$ and $y_{2}$ for plans $A$ and $B$ are both functions of $x$. Please find the analytical expressions for $y_{1}$ and $y_{2}$ separately.  
$(2)$ If plan $A$ is the most cost-effective option, find the range of values for the monthly call time $x$.  
$(3)$ Xiao Ming and Xiao Hua both had a call fee of $60$ yuan in October this year, but Xiao Ming talked longer than Xiao Hua. Find out how much longer Xiao Ming talked compared to Xiao Hua.

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