
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 $f(x) = \begin{cases} \ln x, & x \geq 1 \\ 1- \frac{x}{2}, & x < 1 \end{cases}$, determine the range of $x_1 + x_2$ for the equation $F(x) = f[f(x) + 1] + m$ when it has two zeros $x_1$ and $x_2$.

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