
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
Consider the equation in \( x \):
$$
(1+a) x^{4} + x^{3} - (3a+2) x^{2} - 4a = 0
$$
(1) For any real value of \( a \), does the equation always have a real solution?

(2) Does there exist a real number \( x_{0} \) such that regardless of the real value of \( a \), \( x_{0} \) is never a solution to this equation?

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