
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
For the function $f(x)$ defined for all real numbers $x$ and $y$, the relationship $f(x+y) - f(y) = (x+2y+1)x$ holds, and it's given that $f(1) = 0$.
(Ⅰ) Determine the value of $f(0)$;
(Ⅱ) Find the explicit formula for the function $f(x)$;
(Ⅲ) For any $x_1 \in (0, \frac{1}{2})$ and $x_2 \in (0, \frac{1}{2})$, the inequality $f(x_1) + 2 < \log_a x_2$ is satisfied. Determine the range of possible values for $a$.

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