
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
Given that the listed price of an item is more than $\textdollar 120$, with Coupon A offering $20\%$ off the listed price, Coupon B offering a flat $\textdollar 40$ off, and Coupon C giving $30\%$ off the amount by which the listed price exceeds $\textdollar 120$ plus an additional $\textdollar 20$, determine the difference between the smallest and largest prices at which Coupon A is at least as good as Coupon B or C.

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