
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 the right triangular prism \( ABC-A_1B_1C_1 \), the base is an isosceles right triangle with \(\angle ACB = 90^\circ\). The lateral edge \( AA_1 = 2 \). Points \( D \) and \( E \) are the midpoints of \( CC_1 \) and \( A_1B \), respectively. The projection of point \( E \) onto the plane \( ABD \) is the centroid \( G \) of \(\triangle ABD\). Determine the angle between \( A_1B \) and the plane \( ABD \).

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