
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 about $x$, $|x-2| < a$ ($a \in \mathbb{R}$), be $A$, and $\frac{3}{2} \in A$, $-\frac{1}{2} \notin A$.  
(1) For any $x \in \mathbb{R}$, the inequality $|x-1| + |x-3| \geq a^2 + a$ always holds true, and $a \in \mathbb{N}$. Find the value of $a$.  
(2) If $a + b = 1$, and $a, b \in \mathbb{R}^+$, find the minimum value of $\frac{1}{3b} + \frac{b}{a}$, and indicate the value of $a$ when the minimum is attained.

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