
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
There is a vending machine with 9 toys, each priced between $0.50 and $3.00, with each toy being $0.50 more expensive than the next most expensive toy. Sam presses a button on the machine, which randomly selects one of the remaining toys and offers it for sale. Sam, who has 10 half-dollar coins and a twenty-dollar bill (and the machine only accepts half-dollar coins), will buy the toy if he has enough coins. What is the probability that Sam needs to break the twenty-dollar bill to buy his favorite toy, which costs $2.50? Express your answer as a common fraction.

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