
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 $f(x) = ax^2 + bx + c (a \neq 0)$ whose graph passes through the points $(-2, 0)$ and the origin, and for any $x \in \mathbb{R}$, it satisfies $f(x) \geqslant 2x$.  
$(1)$ Find the expression for the function $f(x)$.  
$(2)$ Let $g(x) = m(x-1)$. If the inequality $f(x) \geqslant g(x)$ holds for $x \in [1, +\infty)$, find the maximum value of the real number $m$.

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