
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
A shopper plans to buy an item priced over $\textdollar 120$. The store offers three discount coupons. Coupon A provides $20\%$ off the total price, Coupon B provides $\textdollar 35$ off the total price, and Coupon C provides $30\%$ off the amount by which the price exceeds $\textdollar 120$. Determine the smallest and largest prices, $x$ and $y$ respectively, at which Coupon A offers at least as much savings as the other two coupons. Find the value of $y - x$.
A) $\textdollar 165$
B) $\textdollar 175$
C) $\textdollar 185$
D) $\textdollar 195$

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