
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 an arithmetic sequence $\{a_n\}$ with a common difference of $2$, the sum of its first $n$ terms is $S_n=pn^2+2n$ $(n\in N^*, p\in R)$.
$(1)$ Find the value of $p$ and the explicit formula for $\{a_n\}$;
$(2)$ In the geometric sequence $\{b_n\}$, $b_2=a_1$, $b_3=a_2+4$, let $c_n= \begin{cases} a_n & (n=2k-1) \\ b_n & (n=2k)\end{cases}$ $(k\in N^*)$, find the sum of the first $n$ terms $T_n$ for the sequence $\{c_n\}$.

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