
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 a sequence $\{a_n\}$ where each term is positive, and the sum of the first $n$ terms is $S_n$, satisfying $a_{n+1}^2=2S_n+n+4$, $a_2-1$, $a_3$, $a_7$ are exactly the first three terms of the geometric sequence $\{b_n\}$.

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

$(2)$ If $C_n=(-1)^n a_n b_n$, 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}
user