
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
Xiao Ming's father is a salesman who uses the phone frequently. Therefore, he chose $A$ types of phone plans at a certain phone service center, as shown in the table below ($A$ and $B$ plans have free incoming calls and charge for outgoing calls, all call durations are for outgoing calls):

| Plan | Fixed Cost (元) | Free Call Time (minutes) | Price per Minute after Free Time (元/minute) |
|------|-----------------|-------------------------|--------------------------------------------|
| $A$  | $18$            | $1500$                  | $0.1$                                      |
| $B$  | $38$            | $4000$                  | $0.07$                                     |

$(1)$ If Xiao Ming's father made phone calls for $2600$ minutes in October, find the phone bill for that month.  
$(2)$ Xiao Ming's father changed to the $B$ plan in November, but the service center informed him that the change would take effect the following month, so he was still billed according to the $A$ plan in November. In January of the next year, Xiao Ming's father found out that the total phone bill for November and December of the previous year was $176$ yuan, and the total call duration was $5200$ minutes. What were the phone bills for November and December?  
$(3)$ If Xiao Ming's father did not change the plan as described in $(2)$, how much more money would he have spent in total for November and December compared to the actual amount?

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