
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 $f(x)=x^{3}$.  
$(1)$ If the point $(1,1)$ lies on the curve $h(x)=bf(x)+cx^{2}+\frac{5}{3}$ and the point $(1,1)$ is the center of symmetry of the graph of the function $h(x)$, find the equation of the tangent line to $h(x)$ passing through the point $(c,b)$.  
$(2)$ If $|f(x)|=a^{x}$ (where $a>0$ and $a\neq 1$) has three distinct zeros $x_{1}$, $x_{2}$, $x_{3}$ with $x_{1} < x_{2} < 0 < x_{3}$, find the range of values for $a$.

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