
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
Given triangle $ABC$ in a plane with two points $P, Q$ that satisfy $\overrightarrow{PA} + \overrightarrow{PC} = \overrightarrow{0}$ and $\overrightarrow{QA} + \overrightarrow{QB} + \overrightarrow{QC} = \overrightarrow{BC}$, and given $|\overrightarrow{AB}| = 4$, $|\overrightarrow{AC}| = 2$, and $S_{\Delta APQ} = \frac{2}{3}$, calculate the value of $\overrightarrow{AB} \cdot \overrightarrow{AC} + {\overrightarrow{BC}}^{2}$.

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