
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 a quadratic function $f(x)$ with a minimum value of 1, and $f(0) = f(2) = 3$, let $g(x) = f(x) + ax$ ($a \in \mathbb{R}$).  
① Find the explicit formula for $f(x)$.  
② If the function $g(x)$ is not monotonic on the interval $[-1, 1]$, find the range of values for the real number $a$.

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