
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
In sequence $\{a_{n}\}$, given that $a_{1}=\frac{1}{2}$, for any $p$, $q\in N^{*}$, we have $a_{p+q}=a_{p}+a_{q}$, let ${b_n}=\frac{1}{{{a_n}{a_{n+1}}}$. Function $f\left(x\right)$ satisfies $f\left(x\right)+f\left(1-x\right)=1$ for any $x\in R$. The sequence $\{a_{n}\}$ satisfies ${a_n}=f(0)+f({\frac{1}{n}})+f({\frac{2}{n}})+…+f({\frac{{n-1}}{n}})+f(1)$, let ${b_n}=\frac{1}{{({{a_n}-\frac{1}{2}})({{a_{n+1}}-\frac{1}{2}})}}$. Choose one of the conditions in ① and ② as the premise, and solve the following problems (Note: If multiple conditions are selected and answered separately, the first answer will be scored):
$(1)$ Is the sequence $\{a_{n}\}$ an arithmetic sequence? Please provide a proof.
$(2)$ 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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