
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 two sets
$$
\begin{array}{l}
A=\{(x, y) \mid |x|+|y|=a, a>0\}, \\
B=\{(x, y) \mid |xy|+1=|x|+|y|\}.
\end{array}
$$

If \( A \cap B \) is the set of vertices of a regular octagon in the plane, determine the value of \( a \).

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