
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 standard six-faced die is rolled twice. Let $a$ be the number shown on the upper face of the die after the first roll and $b$ be the number shown on the upper face of the die after the second roll. Find the probability that the function $y=ax^2-2bx+1$ is a decreasing function on the interval $(-\infty, \frac{1}{2}]$.

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