
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 points $A_{n}(a_{n}, b_{n})$ ($n\in \mathbb{N}, n\geqslant 1$) are distinct points on the graph of the function $y=x^{3}$. Let the first term be $a_{1}=a$ (where $a$ is a constant, $a\in \mathbb{R}, a\geqslant 0$), and define $c_{n}=a_{n}+a_{n+1}$ ($n\in \mathbb{N}, n\geqslant 1$).
- $(1)$ If the sequence $\{c_{n}\}$ is a 5-term geometric sequence with $c_{2}=4$ and $c_{5}=32$, when $a=1$, find the first 6 terms of the sequence $\{a_{n}\}$.
- $(2)$ If the sequence $\{c_{n}\}$ is an infinite arithmetic sequence with $c_{1}=1$ and $c_{3}=7$, when $a=0$, find the sum of the first $n$ terms of the sequence $\{a_{n}\}$.
- $(3)$ If for any $n\in \mathbb{N}$, $n\geqslant 1$, we have $c_{n}=\frac{1}{2}a_{n}+\frac{3}{2}$, and all terms of the sequence $\{a_{n}\}$ are not equal to $1$, let $k_{n}=\frac{b_{n}-1}{a_{n}-1}$. If there exists a constant $k_{0}$ such that for any $n\in \mathbb{N}$, $n\geqslant 1$, the inequality $(k_{n}-k_{0})(k_{n+1}-k_{0}) \lt 0$ holds, find the range of non-negative real number $a$.

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