
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 points $P_1$, $P_2$, $P_3$, and $P_4$ lie on the parabola $y^2 = 8x$ with respective $x$-coordinates $x_1$, $x_2$, $x_3$, and $x_4$, and $F$ is the focus of the parabola $C$, find the value of $|P_1 F| + |P_2 F| + |P_3 F| + |P_4 F|$ given that $x_1 + x_2 + x_3 + x_4 = 10$.

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