
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 triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. The area of the triangle is $S=4\sqrt{3}$, $\angle B=60^{\circ}$, and $a^{2}+c^{2}=2b^{2}$. Consider an arithmetic sequence $\{a_{n}\}$ with $a_{1}=a$ and common difference $d=b$. Let $T_{n}$ be the sum of the first $n$ terms of another sequence $\{b_{n}\}$ such that $T_{n}-2b_{n}+3=0$, where $n\in\mathbb{N}^{\ast}$.

(1) Find the general term formulas for the sequences $\{a_{n}\}$ and $\{b_{n}\}$.
(2) Define a new sequence $\{c_{n}\}$ such that $c_{n}=\begin{cases}a_{n}, & n\text{ is odd}\\b_{n}, & n\text{ is even}\end{cases}$. Find the sum of the first $2n+1$ terms, $P_{2n+1}$.

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