
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 that the minimum value of the quadratic function $f\left(x\right)$ is $1$, and $f\left(0\right)=f\left(2\right)=3$.  
$(1)$ Find the analytical expression of the function $f\left(x\right)$;  
$(2)$ Find the maximum value of $f\left(x\right)$ on the interval $[-\frac{1}{2},\frac{3}{2}]$.

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