
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 copper cube with an edge length of $l = 5 \text{ cm}$ is heated to a temperature of $t_{1} = 100^{\circ} \text{C}$. Then, it is placed on ice, which has a temperature of $t_{2} = 0^{\circ} \text{C}$. Determine the maximum depth the cube can sink into the ice. The specific heat capacity of copper is $c_{\text{s}} = 400 \text{ J/(kg}\cdot { }^{\circ} \text{C})$, the latent heat of fusion of ice is $\lambda = 3.3 \times 10^{5} \text{ J/kg}$, the density of copper is $\rho_{m} = 8900 \text{ kg/m}^3$, and the density of ice is $\rho_{n} = 900 \text{ kg/m}^3$. (10 points)

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