
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
(Ⅰ) The function $f(x)$ satisfies $f(xy) = f(x) + f(y)$ for any real numbers $x$ and $y$, and $f(4) = 2$. Find the value of $f(\sqrt{2})$;  
(Ⅱ) Given that the function $f(x)$ is an odd function defined on the interval \([-1, 1]\) and is increasing on \([-1, 1]\), find the solution set of the inequality $f(x+ \frac{1}{2}) + f(x-1) < 0$.

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