Suppose \( P(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0 \) where \( a_0, a_1, \ldots, a_{n-1} \) are real numbers for \( n \geq 1 \) (monic \( n \)th-degree polynomial with real coefficients). If the inequality
\[ 3(P(x)+P(y)) \geq P(x+y) \] holds for all real numbers \( x,y \), determine the minimum possible value of \( P(2024) \).

Let's think step by step and output the final answer within \boxed{}.
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