
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 $\triangle ABC$, $\angle BAC = \frac{π}{3}$, $D$ is the midpoint of $AB$, $P$ is a point on segment $CD$, and satisfies $\overrightarrow{AP} = t\overrightarrow{AC} + \frac{1}{3}\overrightarrow{AB}$. If $|\overrightarrow{BC}| = \sqrt{6}$, then the maximum value of $|\overrightarrow{AP}|$ is ______.

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