
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
Define a function $f(x)$ over $\mathbb{R}$ that satisfies:
1. For any $x, y \in \mathbb{R}$, $f(x+y) = f(x) + f(y) - 1$;
2. When $x < 0$, $f(x) > 1$.  
(i) Determine the parity of function $f(x) - 1$;  
(ii) Determine the monotonicity of function $f(x)$;  
(iii) If the solution set of the inequality $f(a^2-2a-7) + \frac{1}{2} > 0$ is $\{a \mid -2 < a < 4\}$, find the value of $f(5)$.

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