
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 that $f(x)$ is an even function and $g(x)$ is an odd function, and $f(x) + g(x) = 2^{1-x}$.  
$(1)$ Find the analytical expressions for $f(x)$ and $g(x)$.  
$(2)$ If for any $x \in \mathbb{R}$, there exists $y \in \mathbb{R}$ such that $2^{-y^{2}-2y+n}f(x) \geq 1$ holds, find the range of values for $n$.

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