
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 quadratic function $f(x)$ has a minimum value of 1, and $f(0) = f(2) = 3$.
1. Find the expression for $f(x)$.
2. If $f(x)$ is not monotonic in the interval $[2a, a+1]$, find the range of the real number $a$.
3. In the interval $[-3, 0]$, the graph of $y=f(x)$ is always above the graph of $y=2x+2m+1$. Determine the range of the real number $m$.

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