
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 $f(x)=\frac{2}{3}{x}^{3}-\frac{1}{2}{x}^{2}+(a-1)x+1(a∈R)$.  
$(1)$ If the tangent line of the graph of $f(x)$ at $(1,f(1))$ is perpendicular to the line $x+2y+1=0$, find the value of the real number $a$.  
$(2)$ Discuss the monotonicity of $f(x)$ on the interval $[2,+\infty)$.

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