
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
The side of the base $ABC$ of a regular triangular prism $ABC A_1 B_1 C_1$ is equal to $a$. Points $M$ and $N$ are the midpoints of edges $A_1 B_1$ and $AA_1$, respectively. The projection of segment $BM$ onto line $C_1 N$ is equal to $a / 2 \sqrt{5}$. Determine the height of the prism.

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