
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 arithmetic sequence $\left\{ {{a}\_{n}} \right\}$ has a common difference of $d$, and the geometric sequence $\left\{ {{b}\_{n}} \right\}$ has a common ratio of $q$. Let the sum of the first $n$ terms of $\left\{ {{a}\_{n}} \right\}$ and $\left\{ {{b}\_{n}} \right\}$ be ${{S}\_{n}}$ and ${{T}\_{n}}$ respectively. If ${{n}^{2}}({{T}\_{n}}+1)={{2}^{n}}{{S}\_{n}}$, where $n \in \mathbb{N}^*$, find the values of $d$ and $q$.

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