
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
Let the maximum number among the real numbers $x_1, x_2, \ldots, x_n$ be $\max\{x_1, x_2, \ldots, x_n\}$, and the minimum number be $\min\{x_1, x_2, \ldots, x_n\}$. Given real numbers $1 \leq x \leq y$ and that the three numbers can form the sides of a triangle, if $t = \max\left\{ \frac{1}{x}, \frac{x}{y}, y \right\} \cdot \min\left\{ \frac{1}{x}, \frac{x}{y}, y \right\}$, then the range of $t$ is \_\_\_\_\_\_.

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