
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  $g:[0,1]\rightarrow \mathbb{R}$  be a continuous function and let  $f_{n}:[0,1]\rightarrow \mathbb{R}$  be a 
sequence of functions defined by  $f_{0}(x)=g(x)$  and  $$ f_{n+1}(x)=\frac{1}{x}\int_{0}^{x}f_{n}(t)dt. $$  Determine  $\lim_{n\to \infty}f_{n}(x)$  for every  $x\in (0,1]$ .

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