
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 the 10 integers 0, 1, 2, ..., 9, randomly select 3 different numbers as the coefficients of the quadratic function $f(x)=ax^{2}+bx+c$. The probability that $\frac{f(1)}{2}\in Z$ is \_\_\_\_\_\_.

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