
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  $f(x)=\frac{1+\cos(2 \pi x)}{2}$ , for  $x \in \mathbb{R}$ , and  $f^n=\underbrace{ f \circ \cdots \circ f}_{n}$ . Is it true that for Lebesgue almost every  $x$ ,  $\lim_{n \to \infty} f^n(x)=1$ ?

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