
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
In a unit cube \( ABCD-A_1B_1C_1D_1 \), point \( E \) is on edge \( AA_1 \) such that \( AE = \frac{3 - \sqrt{6}}{3} \). A plane \(\alpha\) is formed through points \( E \), \( C \), and \( D_1 \). Find the dihedral angle between plane \(\alpha\) and face \( DCC_1D_1 \).

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