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

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