
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
Let $f(x) = x^2 - ax + b$.  
$(1)$ If the solution set of the inequality $f(x) < 0$ is $\{x | 2 < x < 3\}$, find the solution set of the inequality $bx^2 - ax + 1 < 0$.  
$(2)$ If $b = 2a - 3$ and $f(x) \geq 0$ always holds, find the range of real numbers for $a$.

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