
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 exponential sequence $\{a_{n}\}$ with a positive common ratio $q$, and the product of the first $n$ terms is $T_{n}$. It is known that $0 \lt a_{1} \lt 1$, and $(a_{50}-1)(a_{49}-1) \lt 0$. If $T_{k}\leqslant T_{n}$ holds for any $n\in N^{*}$, find the value of $k$.

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