Let $a$ and $b$ be real constants such that
\[x^4 + ax^3 + 3x^2 + bx + 1 \ge 0\]for all real numbers $x.$  Find the largest possible value of $a^2 + b^2.$
<|im_start|>system
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<|im_start|>user
Let $a$ and $b$ be real constants such that
\[x^4 + ax^3 + 3x^2 + bx + 1 \ge 0\]for all real numbers $x.$  Find the largest possible value of $a^2 + b^2.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
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Let $a$ and $b$ be real constants such that
\[x^4 + ax^3 + 3x^2 + bx + 1 \ge 0\]for all real numbers $x.$  Find the largest possible value of $a^2 + b^2.$
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