
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 a store offers three different coupons for purchases that exceed $\textdollar 100$, find the smallest and largest possible prices, denoted as $a$ and $b$ respectively, for which Coupon X offers a discount at least as much as the best discount offered by either Coupon Y or Z, and calculate $b - a$.

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