
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 $f(x)=\frac{1}{3}x^{3}-\frac{1}{2}(a-1)x^{2}+bx+1\ (a,b$ are constants, $a > 0)$, the tangent line of the curve $y=f(x)$ at point $P(-1,f(-1))$ is perpendicular to the $y$-axis.

(1) Find the relationship between $a$ and $b$.
(2) Find the extreme values of $f(x)$ on $(0,+\infty)$.

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