
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 $f(x)$ such that for any real numbers $x$ and $y$, it satisfies $f(x+y)+2=f(x)+f(y)$, and for $x>0$, we have $f(x)>2$. Given that $f(3)=5$, find the solution set for the inequality $f(a^2-2a-2)<3$.

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