
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 value of \( x \) that satisfies the equation \( 4y^2 + 2y + 5 = 3(5x^2 + y + 3) \) where \( y = x^2 + 1 \).
A) \( x = \sqrt{1 + \frac{\sqrt{5}}{2}} \) only
B) \( x = \sqrt{1 - \frac{\sqrt{5}}{2}} \) only
C) \( x = \pm \sqrt{1 + \frac{\sqrt{5}}{2}}, \pm \sqrt{1 - \frac{\sqrt{5}}{2}} \)
D) \( x = 0 \) only
E) No real solutions

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