
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 $f(x)$ is an even function and its graph passes through the origin, and the graph of its derivative $f'(x)$ passes through the point $(1,2)$.
$(1)$ Find the analytical expression of the function $f(x)$;
$(2)$ Let $g(x) = f(x) + |f'(x) - m|$, where $m$ is a constant, find the minimum value of the function $g(x)$.

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