
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 $\triangle ABC$, points $D$, $E$, and $F$ are located on $BC$, $CA$, and $AB$ respectively, such that $CD = \frac{3}{5} BC$, $EC = \frac{1}{2} AC$, and $AF = \frac{1}{3} AB$. Let $P$ be a point inside the quadrilateral $AEDF$ (not on the boundary). If $\overrightarrow{DP} = -\frac{1}{3} \overrightarrow{DC} + x \overrightarrow{DE}$, find the range of the real number $x$.

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