
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 positive real numbers $a$ and $b$ satisfying: $a+b=2$, let the minimum value of $\frac{1}{a}+\frac{1}{b}$ be denoted by $m$. Let the function $f(x)=|x-t|+|x+\frac{1}{t}|$ where $(t\ne 0)$. If there exists a real number $x$ such that $f(x)=m$, determine the range of $x$.

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