
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
There are 4 identical particles with a total energy of $4 \mathrm{E}_{0}$, where $\mathrm{E}_{0}$ is a constant. Each particle's energy level can be $k \mathrm{E}_{0}$, where $k=0,1,2,3,4$.

1. If the particles follow Bose-Einstein distribution, where a particle in energy level $k \mathrm{E}_{0}$ can have $k^{2} + 1$ different states, and multiple particles are allowed to occupy the same state at the same energy level, how many distinct distribution diagrams are possible?

2. If the particles follow Fermi-Dirac distribution, where a particle in energy level $k \mathrm{E}_{0}$ has $2\left(1 + k^{2}\right)$ different states, but no two particles can occupy the same state at the same energy level, how many distinct distribution diagrams are possible?

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