
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
(1) Given that $f(x) = (1-3x)(1+x)^5 = a_0 + a_1x + a_2x^2 + \dots + a_6x^6$, find the value of $a_0 + \frac{1}{3}a_1 + \frac{1}{3^2}a_2 + \dots + \frac{1}{3^6}a_6$.  
(2) It is known that in the expansion of $(\sqrt[3]{x} - \frac{1}{2\sqrt[3]{x}})^n$, the sixth term is a constant term. Find the coefficient of the term containing $x^2$.  
(3) Compute the sum $S_{10} = C_{10}^{1} + 2C_{10}^{2} + 3C_{10}^{3} + \dots + 10C_{10}^{10}$.

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