
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 $f_{1}(x)=\sin x$, $f_{n+1}(x)=f_{n}(x) \cdot f_{n}'(x)$, where $f_{n}'(x)$ is the derivative of $f_{n}(x)$ $(n \in \mathbb{N}^{*})$, let the smallest positive period of the function $f_{n}(x)$ be $T_{n}$  
$(1)T_{n}=$_______；  
$(2)$ If $T_{1}+T_{2}+T_{3}+\ldots+T_{n} < k$ always holds, then the minimum value of the real number $k$ is_______．

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