
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 the function $f(x)$ for any $s, t \in \mathbf{R}$ satisfies $f(s+t) - f(t) = (s+2t+1)s$, and $f(1) = 0$. Also, when $x \in \left(0, \frac{1}{2}\right)$, the inequality $f(x) < \log_{a} x - 2$ always holds. Determine the range of the real number $a$.

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