
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
Given the quadratic function $f(x)=x^{2}-x+k$, where $k\in\mathbb{Z}$, if the function $g(x)=f(x)-2$ has two distinct zeros in the interval $(-1, \frac{3}{2})$, find the minimum value of $\frac{[f(x)]^{2}+2}{f(x)}$.

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