
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 \\(\{a_n\}\) is a geometric sequence satisfying \\(a_1=2\), and \\(a_2\), \\(a_3+2\), \\(a_4\) form an arithmetic sequence. The sequence \\(\{b_n\}\) satisfies \\(b_1+ \frac{1}{2}b_2+ \frac{1}{3}b_3+\ldots+ \frac{1}{n}b_n=2n\) for all natural numbers \\(n\).
\\((1)\\) Find the general formula for \\(\{a_n\}\) and \\(\{b_n\}\).
\\((2)\\) Let \\(c_n=(-1)^n(a_n-b_n)\), calculate the sum of the first \\(2n\\) terms, \\(S_{2n}\), of the sequence \\(\{c_n\}\).

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