
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 the Cartesian coordinate plane, given points $A(0,2)$, $B(0,1)$, and $D(t,0)$ with $t > 0$, let $M$ be a moving point on line segment $AD$. If $|AM| \leqslant 2|BM|$ always holds, determine the range of values for the real number $t$.

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