
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 $f(x)=ax^{3}-x^{2}-x+b$, where $(a,b\in \mathbb{R},a\neq 0)$, and $g(x)=\frac{3\sqrt{e}}{4}e^{x}$, the tangent line to the graph of $f(x)$ at $x=-\frac{1}{2}$ is $y=\frac{3}{4}x+\frac{9}{8}$.

$(1)$ Find the values of $a$ and $b$;

$(2)$ Can the line $y=\frac{3}{4}x+\frac{9}{8}$ be tangent to the graph of $g(x)$? If so, write down the coordinates of the tangent point; otherwise, explain why.

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