
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
Given the set A of solutions to the inequality $|x-2| < a$ where $a \in \mathbb{R}$, and it is known that $\frac{3}{2} \in A$ and $-\frac{1}{2} \notin A$.
(1) Find the value of $a \in \mathbb{N}$ that makes the inequality $|x-1| + |x-3| \geq a^2 + a$ always true for all $x \in \mathbb{R}$.
(2) If $a + b = 1$, find the minimum value of $\frac{1}{3|b|} + \frac{|b|}{a}$ and the corresponding value of $a$.

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