
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 the set $M_n=\{n\in\mathbb{N}^*|S= \sum_{i=1}^{n} |i_{2n-1}i_{2n}|\}$ (where $i_1, i_2, \ldots, i_{2n}$ is a permutation of 1, 2, ..., 2n), denote the number of elements in the set $M_n$ as $d_{M_{n}}$. For example, when $n=1$, $M_1=\{1\}$, $d_{M_{1}}=1$; when $n=2$, $M_2=\{2, 4\}$, $d_{M_{2}}=2$; when $n=3$, $M_3=\{3, 5, 7, 9\}$, $d_{M_{3}}=4$.
(1) $M_4$= ______;
(2) Inductively, we have $d_{M_{n}}$= ______.

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