
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 solution set of the inequality $|x-2| < a$ ($a\in\mathbb{R}$) be $A$, and $\frac{3}{2}\in A$, $-\frac{1}{2}\notin A$.  
$(1)$ For any $x\in\mathbb{R}$, $|x-1|+|x-3|\geqslant a^{2}+a$ always holds, and $a\in\mathbb{N}$, find the value of $a$.  
$(2)$ If $a+b=1$, $a$, $b\in\mathbb{R}^{+}$, find the minimum value of $\frac{1}{3b}+ \frac{b}{a}$, and specify the value of $a$ when the minimum value is achieved.

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