
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 \( A^2 + B^2 + C^2 = AB + BC + CA = 3 \) and \( a = A^2 \), find the value of \( a \).

Given that \( n \) and \( b \) are integers satisfying the equation \( 29n + 42b = a \). If \( 5 < b < 10 \), find the value of \( b \).

If \(\frac{\sqrt{3}-\sqrt{5}+\sqrt{7}}{\sqrt{3}+\sqrt{5}+\sqrt{7}}=\frac{c\sqrt{21}-18\sqrt{15}-2\sqrt{35}+b}{59}\), find the value of \( c \).

If \( c \) has \( d \) positive factors, find the value of \( d \).

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