
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 $\triangle ABC$, it is known that $\overrightarrow{CD}=2\overrightarrow{DB}$, $P$ is a point on segment $AD$, and satisfies $\overrightarrow{CP}=\frac{1}{2}\overrightarrow{CA}+m\overrightarrow{CB}$. If the area of $\triangle ABC$ is $\sqrt{3}$ and $∠ACB=\frac{π}{3}$, then the minimum value of the length of segment $CP$ is ______.

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