
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 functions $f(x)=2x^{3}-3x+1$, $g(x)=kx+1-\ln x$.
$(1)$ Let the function $h(x)= \begin{cases} f(x), & x < 1 \\ g(x), & x\geqslant 1 \end{cases}$, discuss the number of zeros of $h(x)$ when $k < 0$;
$(2)$ If there are exactly three tangents passing through point $P(a,-4)$ that are tangent to the curve $y=f(x)$, find the range of values for $a$.

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