
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 $f(x)$ for any $x, y \in \mathbb{R}$, satisfying the condition $f(x) + f(y) = 2 + f(x+y)$, and when $x > 0$, $f(x) > 2$, $f(3) = 5$.  
(1) Determine the monotonicity of the function $f(x)$.  
(2) Find the solution set of the inequality $f(a^2 - 2a - 2) < 3$.

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