
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 equilateral triangle $ABC$ with side length $1$, point $G$ is the midpoint of side $BC$, and points $D$ and $E$ are on line segments $AB$ and $AC$, respectively, with $F$ being the midpoint of $DE$. If $\vec{AD} = \lambda \vec{AB}$ and $\vec{AE} = (1-2\lambda) \vec{AC}$ ($\lambda \in \mathbb{R}$), then the range of $|\vec{FG}|$ is __________.

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