
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 that the constant $a$ is a positive integer, let the set $A=\{x \mid |x-a|<a+\frac{1}{2}, x\in\mathbb{Z}\}$ and the set $B=\{x \mid |x|<2a, x\in\mathbb{Z}\}$. Find the sum of all the elements in the union of sets $A$ and $B$.

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