
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
Determine the distinct digits \(a, b, c, d\) given the following equalities in repeating decimal form:

\[ \cos \alpha = 0.a\dot{b} = 0.abbb\ldots \]
\[ \cos 2\alpha = -0.c\dot{d} = -0.cddd\ldots \]

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