
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 function $f(x)$ with the domain $(0, +\infty)$ that satisfies: ① For any $x \in (0, +\infty)$, it always holds that $f(10x) = 10f(x)$, ② When $x \in (1, 10]$, $f(x) = x - \lg x$, and let the interval $I_k = (10^k, 10^{k+1}]$, where $k \in \mathbb{Z}$, when $x \in I_k$ ($k=0,1,2,3,\ldots$), the values of $f(x)$ form the interval $D_k$. Define the length of the interval $(a, b)$ as $b-a$. Let the length of the complement of interval $D_k$ in interval $I_k$ be $a_k$, then $a_1 = \ $, $a_k = \ $.

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