
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 geometric sequence ${a_n}$ is an increasing sequence, and $a_5^2=a_{10}$, $2(a_n+a_{n+2})=5a_{n+1}$, the sum of the first $n$ terms of the sequence ${b_n}$ is $S_n$, $b_1=1$, $b_n \neq 0$, $b_nb_{n+1}=4S_n-1$.
(1) Find the general term formula for the sequences ${a_n}$ and ${b_n}$;
(2) Let $c_n=a_nb_n$, find the sum of the first $n$ terms of the sequence ${c_n}$, denoted as $T_n$.

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