
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
If $x_{1}$ and $x_{2}$ are two distinct zeros of the derivative of the function $f(x)=\frac{1}{3}{x^3}+a{x^2}+bx+1$ $(a > 0, b > 0)$, and $x_{1}$, $x_{2}$, and $2$ can be arranged to form an arithmetic progression or a geometric progression after proper sorting, calculate the value of $a+b$.

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