
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 a function $y=f(x)$ defined for $(-\infty, +\infty)$, for a given positive number $K$, define the function $f_K(x)$ as follows:
$$
f_K(x) = \begin{cases} 
f(x), & \text{if } f(x) \leq K \\
K, & \text{if } f(x) > K
\end{cases}.
$$
Let $f(x) = 2^{-|x|}$. When $K=\frac{1}{2}$, determine the interval over which the function $f_K(x)$ is monotonically increasing.

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