
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 $\triangle ABC$, point $D$ satisfies $\overrightarrow{DC}=2\overrightarrow{AD}$. If a point $P$ on segment $BD$ satisfies $\overrightarrow{AP}=x\overrightarrow{AB}+y\overrightarrow{AC}$ ($x>0$, $y>0$), then the range of $y-x$ is ______.

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