
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 $f(x) = |x - \frac{3}{2}| - a$. If the solution set of the inequality $f(x) < 0$ is $M$, and $\frac{1}{2} \in M$, $-\frac{1}{2} \notin M$,
$(1)$ find the maximum value of the real number $a$;
$(2)$ when $a \in \mathbb{N}^*$, if the inequality $|x - a| - |x - 3| > b$ has a solution, find the range of values for the real number $b$.

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