
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
To celebrate Mother's Day, a shopping mall launched a promotion of giving out "coupons" to stimulate consumption. There are three types of coupons as shown in the table below:

| $A$ Type | $B$ Type | $C$ Type |
|----------|----------|----------|
| Spend $368$ get $100$ off | Spend $168$ get $68$ off | Spend $50$ get $20$ off |

In this promotion, Xiao Wen received several coupons of three different types and planned to buy a gift for her mother.

$(1)$ If Xiao Wen uses three different types of coupons at the same time and saves a total of $520$, and it is known that she used $1$ $A$ Type coupon and $4$ $C$ Type coupons, then how many $B$ Type coupons did she use?

$(2)$ If Xiao Wen used $5$ $A$ and $B$ Type coupons at the same time and saved a total of $404$, how many $A$ and $B$ Type coupons did she use?

$(3)$ If Xiao Wen received a total of $16$ coupons for each of the three different types of coupons (some were not used), and she used two different types of coupons from $A$, $B$, and $C$ Type at the same time, saving a total of $708$, what are the possible combinations of coupons used? (Please provide the specific solution process)

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