
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 a sequence of real numbers $\{a_n\}$ that satisfies: $a_{n+2} = |a_{n+1}| - a_n$ $(n=1,2,\cdots)$, where $a_1 = a$, $a_2 = b$, let set $M = \{a_n | n \in \mathbb{N}^*\}$.

(I) If $a = 1$, $b = 2$, write out set $M$ using the listing method;

(II) If $a < 0$, $b < 0$, determine whether the sequence $\{a_n\}$ is a periodic sequence, and explain the reason;

(III) If $a \geqslant 0$, $b \geqslant 0$, and $a + b \neq 0$, find the minimum number of elements in set $M$.

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