
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 the binomial expansion of $(\sqrt{x} - \frac{2}{\sqrt{x}})^n$, the ratio of the binomial coefficient of the fourth term to that of the third term is 8:3.  
(1) Find the value of $n$;  
(2) Find the coefficient of the $x^3$ term in the expansion;  
(3) Calculate the value of $C_{10}^{0} - 2C_{10}^{1} + 4C_{10}^{2} - 8C_{10}^{3} + \ldots + 1024C_{10}^{10}$.

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