
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 sequence $\{a_{n}\}$, if it satisfies $a_{1}=a (a>0, a\neq 1)$, and for any $m$, $n\in N^{*}$, $a_{m+n}=a_{m}\cdot a_{n}$, then $\{a_{n}\}$ is called an "exponential sequence". If the sequence $\{a_{n}\}$ satisfies: $a_{1}=1$, $a_{n}=2a_{n+1}+a_{n}\cdot a_{n+1}$;  
$(1)$ Determine whether $\left\{\frac{1}{{a_n}}+1\right\}$ is an "exponential sequence". If it is, provide a proof; if not, explain the reason;  
$(2)$ If ${b_n}=\frac{1}{{a_n}}+n$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$ denoted as $T_{n}$.

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