
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 G be the centroid of $\triangle ADE$, and let point P be any point within (including the boundary) $\triangle DEG$. Points B and C are the trisection points on AD and AE, respectively (closer to point A). If $\overrightarrow{AP} = \alpha \overrightarrow{AB} + \beta \overrightarrow{AC}$ ($\alpha, \beta \in \mathbb{R}$), determine the range of $\alpha + \frac{1}{2}\beta$.

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