
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 that the odd function \( f(x) \) is increasing on the interval \( (-\infty, 0) \), and that \( f(-2) = -1 \), \( f(1) = 0 \). When \( x_{1} > 0 \) and \( x_{2} > 0 \), it holds that \( f\left( x_{1} x_{2} \right) = f\left( x_{1} \right) + f\left( x_{2} \right) \). Find the solution set for the inequality \( \log _{2}|f(x)+1|<0 \).

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