
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 $f(x)=x^{3}+x$ has a tangent line at $x=1$, denoted as $m$.  
$(1)$ Find the equation of the tangent line $m$;  
$(2)$ If the tangent line of the curve $g(x)=\sin x+ax$ at point $A(0,g(0))$ is perpendicular to $m$, find the value of the real number $a$.

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