
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 values of the parameter \( \boldsymbol{a} \) for which the equation \( (a+1)\left(x^{2}+1\right)^{2}-(2a+3)\left(x^{2}+1\right)x+(a+2)x^{2}=0 \) has exactly two distinct real roots.

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