
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
Show, without using the formula for the sum of a geometric series, that if

$$
\left\{\begin{array}{l}
1+q+q^{2}+\ldots+q^{2 n-1}+q^{2 n}=S \\
1-q+q^{2}-\ldots-q^{2 n-1}+q^{2 n}=S_{1}
\end{array}\right.
$$

then

$$
1+q^{2}+q^{4}+\ldots+q^{4 n-2}+q^{4 n}=S S_{1}
$$

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