
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
A plate is bounded by the parabola \( y^{2} = x \) and its chord passing through the focus perpendicularly to the axis of the parabola. Find the mass of the plate if at each point the surface density is inversely proportional to the distance from the point to the directrix of the parabola.

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