
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 cubic tank with rigid walls, with an edge length of \( a = 1 \) m, is divided by a thin, light, undeformable piston that moves vertically without friction. Initially, the tank is completely filled with water having a mass of \( M = 1000 \) kg. How much neon mass should be pumped into the space under the piston, so that the piston moves a distance of \( \Delta h = 0.5 \) mm? The temperatures of neon and water are the same, constant, and equal to \( t = 32^{\circ} \mathrm{C} \). The molar mass of neon is \( \mu = 20 \) g/mol, the value of the universal gas constant is \( R = 8.31 \) J/(K·mol), and the acceleration due to gravity is \( g = 10 \ \text{m} / \text{c}^{2} \). The compressibility of water (relative volume change under isothermal pressure increase) is \( \varepsilon = 5 \cdot 10^{-10} \ \text{Pa}^{-1} \).

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