
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 $f(x)=\begin{cases} x+k(1-a^{2}), & x\geqslant 0 \\ x^{2}-4x+(3-a)^{2}, & x < 0 \end{cases}$, where $a\in \mathbb{R}$, find the range of values for $k$ such that for any nonzero real number $x_{1}$, there exists a unique nonzero real number $x_{2} (x_{1} \neq x_{2})$, such that $f(x_{1})=f(x_{2})$ holds.

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