
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 function $f\left(x\right)=2x^{2}+bx+c$ ($b$, $c$ are real numbers), $f\left(-10\right)=f\left(12\right)$. If the equation $f\left(x\right)=0$ has two positive real roots $x_{1}$ and $x_{2}$, then the minimum value of $\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}}$ is ______.

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