
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 the odd coefficients in the binomial expansion of $(( \sqrt {x}- \sqrt[3]{x})^{n})$ is $512$,
$(1)$ Find all the rational terms (terms with integer exponents) in the expansion.
$(2)$ Find the coefficient of the $x^{2}$ term in the expansion of $((1-x)^{3}+(1-x)^{4}+…+(1-x)^{n})$.

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