
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 $f(x)=\frac{x}{x+2}$ where $x > 0$, observe the following:

$${f_{1}}(x)=f(x)=\frac{x}{x+2}$$
$$f_{2}(x)=f(f_{1}(x))=\frac{x}{3x+4}$$
$$f_{3}(x)=f(f_{2}(x))=\frac{x}{7x+8}$$
$$f_{4}(x)=f(f_{3}(x))=\frac{x}{15x+16}$$
$$\cdots \cdots$$

Based on the above observations, use mathematical induction to find the expression for ${f_{n}}(x)=f(f_{n-1}(x))$ when $n \in \mathbb{N}^{*}$ and $n \geqslant 2$.

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