
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 constants $a$ and $b$, with $a \neq 0$, and $f(x) = ax^2 + bx$, $f(2) = 0$.

$(1)$ If the function $y = f(x) - x$ has a unique zero point, find the expression for the function $f(x)$;

$(2)$ Find the maximum value of the function $f(x)$ in the interval $[-1,2]$;

$(3)$ When $x \geqslant 2$, the inequality $f(x) \geqslant 2 - a$ always holds, find the range of the real number $a$.

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