
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 $\begin{cases} x+k(1-a^{2}),(x\geqslant 0)\\ x^{2}-4x+(3-a)^{2},(x < 0)\\ \end{cases}$ where $a\in R.$, determine the range of $k$ such that for any non-zero real number $x_{1}$, there exists a unique non-zero 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