
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 function defined on the interval $0 \le x < 1$:

\[ 
f_{1}(x) = \begin{cases} 
1, & \text{if } 0.5 \le x < 0.6 \\ 
0, & \text{otherwise} 
\end{cases} 
\]

determine the elements of the sequence of functions $f_{n}(x)$ step by step using the following procedure. Where $f_{n-1}(x) > 0$, let $f_{n}(x) = f_{n-1}(x)$; where $f_{n-1}(x) = 0$, let $f_{n}(x)$ be $q^{n-1}$ or $0$ according to whether the $n$-th decimal digit of $x$ is 5 or not, where $q$ is a positive number less than 1. Determine the limit of the sequence 

\[
a_{n} = \int_{0}^{1} f_{n}(x) \, dx \quad (n = 1, 2, \ldots) 
\]

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