
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 when $x \in \left(-\frac{1}{2}, \frac{1}{2}\right)$, $\frac{1}{{1+2x}}=1-2x+4{x^2}-\ldots+{(-2x)^n}+\ldots $, if for any $x \in \left(-\frac{1}{2}, \frac{1}{2}\right)$, $\frac{x}{{(1-x^3)(1+2x)}}={a_0}+{a_1}x+\ldots+{a_n}{x^n}+\ldots $, then $a_{9}=$____.

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