
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
Define "n-m" as the "length" of the set $\{x | m \leq x \leq n\}$. Given that $a$ and $b$ are real numbers, let set $A = \{x | a \leq x \leq a+1\}$, and set $B = \{x | b- \frac{1}{2} \leq x \leq b+1\}$, and both $A$ and $B$ are subsets of the set $U = \{x | 1 \leq x \leq 3\}$. Then, the minimum value of the length of $A \cap B$ is __________.

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