
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 that the sum of all odd-numbered terms in the binomial expansion of ${\left(\sqrt{x}-\sqrt[3]{x}\right)}^{n}$ is $512$,

(1) Find all rational terms (terms with integral exponents) in the expansion.

(2) Find the coefficient of the ${x}^{2}$ term in the expansion of ${\left(1-x\right)}^{3}+{\left(1-x\right)}^{4}+\cdots +{\left(1-x\right)}^{n}$.

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