
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 $\mathbb{R}$ with a period of $1$. In the interval $[0,1)$, $f(x)= \begin{cases} x^{2}, & x\in D \\ x, & x\notin D \end{cases}$, where the set $D=\left\{x \mid x= \frac{n-1}{n}, n\in \mathbb{N}^{*}\right\}$. Find the number of solutions to the equation $f(x)-\lg x=0$.

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