
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
Suppose that \( a = \cos^4 \theta - \sin^4 \theta - 2 \cos^2 \theta \), find the value of \( a \).

If \( x^y = 3 \) and \( b = x^{3y} + 10a \), find the value of \( b \).

If there is (are) \( c \) positive integer(s) \( n \) such that \( \frac{n+b}{n-7} \) is also a positive integer, find the value of \( c \).

Suppose that \( d = \log_{4} 2 + \log_{4} 4 + \log_{4} 8 + \ldots + \log_{4} 2^{c} \), find the value of \( d \).

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