
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 length of the base \(ABC\) of a regular prism \(ABC A_1 B_1 C_1\) is \(a\). Points \(M\) and \(N\) are the midpoints of the edges \(AC\) and \(A_1 B_1\), respectively. The projection of segment \(MN\) onto the line \(B A_1\) is \(\frac{a}{2 \sqrt{6}}\). Determine the height of the prism (find all solutions).

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