
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 \( f(x)=\frac{1010 x+1009}{1009 x+1010} \) be defined as:
\[ f^{(1)}(x)=f(x), \]
\[ f^{(i)}(x)=f\left(f^{(i-1)}(x)\right) \quad \text{for } i = 2, 3, \cdots. \]

Find \( f^{(n)}(x) \).

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