
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
Equilateral triangle \( \triangle ABC \) and square \( ABDE \) have a common side \( AB \). The cosine of the dihedral angle \( C-ABD \) is \(\frac{\sqrt{3}}{3}\). If \( M \) and \( N \) are the midpoints of \( AC \) and \( BC \) respectively, then the cosine of the angle between \( EM \) and \( AN \) is \(\qquad\).

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