
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 ${{a}_{n}}$ that satisfies ${a}_{n+2}=q{a}_{n}$ ($q$ is a real number and $q\neq 1$), where $n\in {N}^{*}$, ${a}_{1}=1$, and ${a}_{2}=2$, and ${{a}_{2}}+{{a}_{3}},{{a}_{3}}+{{a}_{4}},{{a}_{4}}+{{a}_{5}}$ forms an arithmetic sequence.

(1) Find the value of $q$ and the general term formula for ${{a}_{n}}$;

(2) Let ${{b}_{n}}=\dfrac{{{\log }_{4}}{{a}_{2n}}}{{{a}_{2n-1}}}$, $n\in {{N}_{+}}$, find the sum of the first $n$ terms of the sequence $\left\{ {{b}_{n}} \right\}$.

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