
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 $f(x)=x^{3}+ax^{2}+x+1$.  
$(1)$ If the tangent line of $y=f(x)$ at the point $(1,f(1))$ is parallel to the line $2x-y+1=0$, find the equation of the tangent line of $y=f(x)$ passing through the point $(0,1)$.  
$(2)$ If the function $f(x)$ is decreasing on the interval $[-\frac{2}{3},-\frac{1}{3}]$, find the range of real numbers for $a$.

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