
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 vending machine contains 10 toys. Each toy costs between 50 cents and 5 dollars, with each toy being 50 cents more expensive than the next most expensive one. Every time Jane presses a green button, the machine randomly selects one of the remaining toys and offers it to Jane. If Jane has enough money, she buys the toy, and can press the button again. If Jane starts with 10 half-dollar coins and a twenty-dollar bill, and the machine only accepts half-dollars, what is the probability that Jane has to get change for the twenty-dollar bill before she can buy her favorite toy, which costs $\$4.50$? Express your answer as a common fraction.

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