
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 $f(x)=\frac{1}{3}x^3-\frac{1}{2}x^2+1$, where $x\in \mathbf{R}$.

(1) Find the maximum and minimum values of the function $f(x)$;

(2) Find the equation of the tangent line that passes through the point $(\frac{3}{2},1)$ on the function graph;

(3) Find the area of the closed shape enclosed by the graph of the function $f(x)=\frac{1}{3}x^3-\frac{1}{2}x^2+1$ and the straight line $y=1$.

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