<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 Let the sequence $(a_n)$ be defined as $a_1 = \\frac{1}{3}$ and $a_{n+1} = \\frac{a_n}{\\sqrt{1 + 13a_n^2}}$ for every $n \\geq 1$. Find the largest value of $k$ such that $a_k < \\frac{1}{50}$. Express your answer as a single integer value for $k$. Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
193
Let the sequence $(a_n)$ be defined as $a_1 = \\frac{1}{3}$ and $a_{n+1} = \\frac{a_n}{\\sqrt{1 + 13a_n^2}}$ for every $n \\geq 1$. Find the largest value of $k$ such that $a_k < \\frac{1}{50}$. Express your answer as a single integer value for $k$.
test