  JWST has brought us new insights into Cosmic Dawn with tentative detection of
the unique signatures of metal-free Population III (Pop III) stars, such as
strong HeII emission, extremely blue UV spectrum, and enhanced nitrogen
abundance. Self-consistent theoretical predictions of the formation rates,
sites, and masses of Pop III stars are crucial for interpreting the
observations, but are challenging due to complex physical processes operating
over the large range of length scales involved. One solution is to combine
analytical models for the small-scale star formation process with cosmological
simulations that capture the large-scale physics such as structure formation,
radiation backgrounds, and baryon-dark matter streaming motion that regulate
the conditions of Pop III star formation. We build an analytical model to
predict the final masses of Pop III stars/clusters from the properties of
star-forming clouds, based on the key results of small-scale star formation
simulations and stellar evolution models. Our model for the first time
considers the interplay between feedback and fragmentation and covers different
modes of Pop III star formation ranging from ordinary small ($\sim 10-2000\ \rm
M_\odot$) clusters in molecular-cooling clouds to massive ($\gtrsim 10^{4}\ \rm
M_\odot$) clusters containing supermassive ($\sim 10^{4}-3\times 10^{5}\ \rm
M_\odot$) stars under violent collapse of atomic-cooling clouds. As an example,
the model is applied to the Pop III star-forming clouds in the progenitors of
typical haloes hosting high-$z$ luminous quasars, which shows that formation of
Pop III massive clusters is common ($\sim 20-70\%$) in such biased
($\sim4\sigma$) regions, and the resulting heavy black hole seeds from
supermassive stars can account for a significant fraction of observed luminous
($\gtrsim 10^{46}\ \rm erg\ s^{-1}$) quasars at $z\sim 6$.
