
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 the function $f(x)=x^{2}+|x-a|+1$, where $x\in \mathbb{R}, a\in \mathbb{R}$.

$(1)$ When $a=1$, find the minimum value of the function $f(x)$;

$(2)$ If the minimum value of the function $f(x)$ is denoted as ${g}(a)$, and let $m={g}(a)$, find the range of $m$.

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