
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 $f(x)=\frac{x}{2x+2} (x>0)$, observe the following:
$f_{1}(x)=f(x)=\frac{x}{2x+2}$,
$f_{2}(x)=f(f_{1}(x))=\frac{x}{6x+4}$,
$f_{3}(x)=f(f_{2}(x))=\frac{x}{14x+8}$,
$f_{4}(x)=f(f_{3}(x))=\frac{x}{30x+16}$
...
Based on these observations, use induction to find the value of $f_{n}(1)=$ _______ when $n \in \mathbb{N}^*$.

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