The Cost of One Mistake

A polynomial \(P(x)\) initially follows the simple linear pattern \[ P(k)=k+1 \qquad (k=0,1,\ldots,2025). \] While entering the next value, a typographical error changes a single digit, causing the system to record \[ P(2026)=20{\color{red}{3}}7 \quad\text{instead of}\quad 20{\color{green}{2}}7. \] Rather than correcting the typo, the programmer decides to keep all the recorded values and redefine \(P(x)\) as a polynomial of degree \(2026\). What value must the new polynomial assign to $P(2027)$?