
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
Let the number of nickels be $n$, the number of dimes be $d$, the number of quarters be $q$, and the number of half-dollars be $h$. If the total number of coins is 120, then we have $n + d + q + h = 120$, and the total value of the coins is $0.05n + 0.10d + 0.25q + 0.50h = 12.40$. Find the difference between the largest and smallest number of dimes that could be in the bank.

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