
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 $f(x)= \begin{cases} -x^{2}-2x+1 &(x\leqslant 0) \\ |\log _{2}x| &(x > 0)\end{cases}$, find the range of $x_{1}+x_{2}+x_{3}+x_{4}$ if the equation $f(x)=k$ has four distinct real roots, $x_{1}$, $x_{2}$, $x_{3}$, and $x_{4}$.

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