
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 $∠BAC=\frac{π}{3}$. Point $D$ is the midpoint of segment $AB$, and point $P$ is a point on segment $CD$ satisfying $\overrightarrow{AP}=t\overrightarrow{AC}+\frac{1}{3}\overrightarrow{AB}$ where $t∈{R}$. If the area of $\triangle ABC$ is $\frac{{3\sqrt{3}}}{2}$, then the minimum value of $|\overrightarrow{AP}|$ is ____.

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