
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 a cubic function $f\left(x\right)=x^{3}+ax^{2}+4x+1$ where $a$ is a constant.<br/>$(1)$ When $a=1$, find the equation of the tangent line to the function $f\left(x\right)$ at $x=2$;<br/>$(2)$ If $a \lt 0$, discuss the monotonicity of the function $f\left(x\right)$ in the interval $x\in \left(0,+\infty \right)$.

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