
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 $f(x) = ax^2 - 2ax + b + 2 (a > 0)$, where the maximum value on the interval $[0,1]$ is $3$ greater than the minimum value, and $f(1) = 0$.  
$(1)$ Find the values of $a$ and $b$.  
$(2)$ For $x \in [\frac{1}{3},2]$, the graph of the function $y = f(x)$ is always below the graph of the function $y = mx^2 + 1$. Find the range of real numbers for $m$.

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