
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 \( k \in \mathbf{Z} \), let \( I_k \) denote the interval \( (2k-1, 2k+1] \). Given that \( f(x) = x^2 \) when \( x \in I_k \):
1. Find the explicit expression of \( f(x) \) on \( I_k \).
2. For positive integers \( k \), find the set \( M_k = \{ a \mid \) the equation \( f(x) = ax \) has two distinct real roots in \( I_k \} \).

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