
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
Let $f(x)$ be a function defined on the interval $(-\infty, +\infty)$ with a period of 2. For $k \in Z$, let $I_{k}$ denote the interval $(2k - 1, 2k + 1]$. It is known that $f(x) = x^2$ when $x \in I_{0}$.

(1) Find the analytic expression of $f(x)$ on $I_{k}$.

(2) For natural numbers $k$, find the set $M_{k} = \{a \mid$ the equation $f(x) = a x$ has two distinct real roots in $I_{k} \}$.

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