
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 $f(x)=x-\ln x+h$ on the interval $\left[\frac{1}{e},e^{2}\right]$, if three real numbers $a$, $b$, $c$ are arbitrarily chosen from this interval, and there always exists a triangle with side lengths $f(a)$, $f(b)$, and $f(c)$, determine the range of the real number $h$.

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