
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 $P_1(x_1, y_1), P_2(x_2, y_2), \ldots, P_{10}(x_{10}, y_{10})$ are on the parabola $(y^2 = 2x)$, and $(x_1 + x_2 + \ldots + x_{10} = 10)$, find the value of $|P_1F| + |P_2F| + \ldots + |P_{10}F|$.

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