
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 $$f(x)=(1-3x)(1+x)^{5}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{6}x^{6}$$, find $$a_{0}+ \frac {1}{3}a_{1}+ \frac {1}{3^{2}}a_{2}+\ldots+ \frac {1}{3^{6}}a_{6}$$.
2. Given that in the expansion of $$\left( \sqrt[3]{x} - \frac {1}{2 \sqrt[3]{x}} \right)^{n}$$, the sixth term is a constant term, find the coefficient of the term containing $x^{2}$.
3. Find the sum $$S_{10}= C_{ 10 }^{ 1 }+2 C_{ 10 }^{ 2 }+3 C_{ 10 }^{ 3 }+\ldots+10 C_{ 10 }^{ 10 }$$.

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