
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
There are three conditions:<br/>① The graph of the function $f(x)$ passes through the point $(0,1)$, and $a=1$;<br/>② $f(x)$ has a maximum value of $\frac{11}{6}$ at $x=1$;<br/>③ The tangent line equation of the function $f(x)$ at $x=3$ is $4x-2y-7=0$. Choose an appropriate condition from the three conditions to complete the following question (only fill in the number), and solve the problem.<br/>Question: Given the function $f(x)=\frac{1}{3}x^3+\frac{a}{2}x^2+2x+b$ has extreme values, and _____.<br/>$(1)$ Find the analytical expression of $f(x)$;<br/>$(2)$ Find the extreme values of the function $f(x)$ when $x\in [1,3]$.

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