
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]


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Given a sequence \\(\{a_n\}\) where each term is positive and satisfies \\(a_{n+1}^2-a_{n+1}a_n-2a_n^2=0\\), for \\(n\in\mathbb{N}^*\), and \\(a_3+2\\) is the arithmetic mean of \\(a_2\\) and \\(a_4\\). Another sequence \\(\{b_n\}\) satisfies \\(b_1=1\\), and \\(b_{n+1}=b_n+2\\).

\\((1)\\) Find the general formula for the sequences \\(\{a_n\}\) and \\(\{b_n\}\).

\\((2)\\) Let \\(c_n= \dfrac{1-(-1)^n}{2}a_n- \dfrac{1+(-1)^n}{2}b_n\\), find the sum of the first \\(2n\\) terms of the sequence \\(\{c_n\}\), denoted as \\(T_{2n}\\).

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