
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
Let $\overrightarrow{P}$ be a point inside $\triangle ABC$ such that $\overrightarrow{AP} + \overrightarrow{BP} + \overrightarrow{CP} = \overrightarrow{0}$ and $\overrightarrow{BD} = \frac{1}{3}\overrightarrow{BC}$. Determine the value of $\overrightarrow{AD} + \overrightarrow{AP}$.

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