We propose CoLyricist, an AI-assisted lyric writing tool designed to support the typical workflows of experienced lyricists and enhance their creative efficiency. While lyricists have unique processes, many follow common stages. Tools that fail to accommodate these stages challenge integration into creative practices. Existing research and tools lack sufficient understanding of these songwriting stages and their associated challenges, resulting in ineffective designs. Through a formative study involving semi-structured interviews with 10 experienced lyricists, we identified four key stages: Theme Setting, Ideation, Drafting Lyrics, and Melody Fitting. CoLyricist addresses these needs by incorporating tailored AI-driven support for each stage, optimizing the lyric writing process to be more seamless and efficient. To examine whether this workflow-aligned design also benefits those without prior experience, we conducted a user study with 16 participants, including both experienced and novice lyricists. Results showed that CoLyricist enhances the songwriting experience across skill levels. Novice users especially appreciated the Melody-Fitting feature, while experienced users valued the Ideation support.