
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
The function $y=mx^{3}+2x+1$ has a tangent line $l$ at the point $\left(1,m+3\right)$.<br/>$(1)$ If $l$ is parallel to the line $y=3x$, find the value of the real number $m$;<br/>$(2)$ If $l$ is perpendicular to the line $y=-\frac{1}{2}x$, find the equation of the tangent line $l$.

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