
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 that the coordinates of the vertices of triangle $\triangle A B C$ are $A(0,0)$, $B(7,0)$, $C(3,4)$, and a line passing through the point $(6-2 \sqrt{2}, 3-\sqrt{2})$ intersects segments $AC$ and $BC$ at points $P$ and $Q$ respectively. If the area of $\triangle P Q C$ is $\frac{14}{3}$, then the value of $|C P| + |C Q| = \quad$.

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