
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 $f(x)=\left\{\begin{array}{l}{-2x+2, x\geq 0}\\{(\frac{1}{3})^{x}+1, x<0}\end{array}\right.$. Find the range of real numbers $x$ that satisfy the inequality $f(x)+f(x+\frac{1}{2}) \gt 2$.

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