
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
If \( a(x+1)=x^{3}+3x^{2}+3x+1 \), find \( a \) in terms of \( x \).

If \( a-1=0 \), then the value of \( x \) is \( 0 \) or \( b \). What is \( b \) ?

If \( p c^{4}=32 \), \( p c=b^{2} \), and \( c \) is positive, what is the value of \( c \) ?

\( P \) is an operation such that \( P(A \cdot B) = P(A) + P(B) \).

\( P(A) = y \) means \( A = 10^{y} \). If \( d = A \cdot B \), \( P(A) = 1 \) and \( P(B) = c \), find \( d \).

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