
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  $f(x)$  be the function defined for  $x\geq 0$  which satisfies the following conditions.
(a)  $f(x)=\begin{cases}x \ \ \ \ \ \ \ \ ( 0\leq x<1)  2-x \ \ \ (1\leq x <2) \end{cases}$  

(b)  $f(x+2n)=f(x) \ (n=1,2,\cdots)$ 

Find  $\lim_{n\to\infty}\int_{0}^{2n}f(x)e^{-x}\ dx.$ 

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