
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
In the geometric sequence $\{a_n\}$, $a_1=512$ and the common ratio $q=-\frac{1}{2}$. Let $\prod_{n}=a_{1}\times a_{2}\times \ldots \times a_{n}$. Determine the number of values among $\prod_{8}$, $\prod_{9}$, $\prod_{10}$, $\prod_{11}$ that are positive.

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