
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 $a < b < 0$, the odd function $f(x)$ is defined on the domain $[a, -a]$, and it is monotonically decreasing on the interval $[-b, -a]$ and $f(x) > 0$. Determine the sign of $f(x)$ and the monotonicity of $|f(x)|$ on the interval $[a, b]$.

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