
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 domain of $y=f(x)$ is $[1,4]$, $f(1)=2$, $f(2)=3$. When $x\in[1,2]$, the graph of $f(x)$ is a line segment; when $x\in[2,4]$, the graph of $f(x)$ is part of a quadratic function graph, and its vertex is $(3,1)$.
(1) Find the analytical expression of $f(x)$;
(2) Find the range of $f(x)$.

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