
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 mobile company offers two phone billing methods, $A$ and $B$. 

| Billing Method | Monthly Fee ($/ \text{元}$) | Outgoing Call Limit ($/\text{min}$) | Overtime Fee for Outgoing Calls ($(\text{元}/\text{min})$) | Incoming Calls |
|----------------|-----------------------------|------------------------------------|-----------------------------------------------------------|----------------|
| $A$            | $78$                        | $200$                              | $0.25$                                                    | Free           |
| $B$            | $108$                       | $500$                              | $0.19$                                                    | Free           |

$(1)$ Suppose the outgoing call time of a mobile phone within a month is $t \text{ min}$. According to the table above, write down the analytical expressions of the billing amounts $y_{1}$ and $y_{2}$ for methods $A$ and $B$ respectively, within different time ranges.  
$(2)$ If you expect to have an outgoing call time of $350 \text{ min}$ per month, which billing method, $A$ or $B$, would you choose and explain the reason?  
$(3)$ Based on different ranges of monthly outgoing call time $t$, directly write down the most cost-effective billing method.

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