
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
From nine slips numbered 1 to 9, we draw three numbers sequentially - without replacement - denoted as \(a, b\), and \(c\). If we form the quadratic equation

\[ 
a x^{2} + b x + c = 0 
\]

what is the probability that
a) In 10 trials, at least once the two roots are equal?
b) In 5 trials, exactly once and only once it occurs that the product of the two roots is a decimal number with a finite number of decimal places and is less than 1?

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