
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 the function $f(x) = x^3 - \frac{9}{2}x^2 + 6x - a$.

$(1)$ For any real number $x$, $f'(x) \geqslant m$ always holds, find the maximum value of $m$;

$(2)$ If the equation $f(x) = 0$ has exactly one real root, find the range of values for $a$.

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