
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
Investigate the convergence of the series

$$
\frac{1}{2} x+\frac{2}{2^{2}} x^{2}+\frac{3}{2^{3}} x^{3}+\frac{4}{2^{4}} x^{4}+\ldots+\frac{n}{2^{n}} x^{n}+\ldots
$$

at the points $x=1, x=3, x=-2$.

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