
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 \( a \) and \( b \) are the roots of \( x^{2}+x \sin \alpha+1=0 \) while \( c \) and \( d \) are the roots of the equation \( x^{2}+x \cos \alpha-1=0 \). Find the value of \( \frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{d^{2}} \).

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