
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 $f(x)= \begin{cases} 2^{x}-1, & x\leqslant 0\\ f(x-1)+1, & x > 0 \end{cases}$, arrange the roots of the equation $f(x)-x=0$ in ascending order to form a sequence. The sum of the first $n$ terms of this sequence, $S_{n}=$ \_\_\_\_\_\_.

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