
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 and his parents plan to go to a hotpot restaurant to eat hotpot. The restaurant sells "$25$ for $50$" universal vouchers online (meaning that you can get a $50$ yuan voucher for only $25$ yuan). The restaurant stipulates that vouchers can be used as cash, with a maximum of $3$ vouchers per transaction, and the voucher amount cannot exceed the total amount to be paid.<br/>$(1)$ If Xiao Ming's family has a total amount to pay of $145$ yuan, how much can they save at most by using vouchers? <br/>$(2)$ After Xiao Ming's family arrives at the hotpot restaurant, they find that there is another discount available on-site: except for the hotpot base, all other dishes are discounted by $40\%$. Xiao Ming's family orders a $50$ yuan hotpot base and other dishes. After dining, clever Xiao Ming compares the two discount options and chooses to pay on-site with the discount, saving $15$ yuan more than using vouchers. How much did Xiao Ming's family actually pay?

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