
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 ticket prices for a certain tourist attraction are as follows:

| Number of Tickets | $1\sim 50$ tickets | $51\sim 100$ tickets | Over 100 tickets |
|-------------------|---------------------|----------------------|------------------|
| Price per ticket   | $15$ yuan           | $12$ yuan            | $10$ yuan        |

A total of $105$ students from the 7th grade classes (1) and (2) went on a trip. In class (1), there were more than $40$ students but less than $50$ students. After calculation, if both classes purchase tickets as a group, the total amount to be paid is $1401$ yuan.

$(1)$ How many students are there in each class?
$(2)$ If the two classes combine and purchase tickets as a group, how much money can be saved?
$(3)$ If the 7th grade class (1) organizes the trip independently, as the organizer, how can you save money when purchasing tickets?

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