  The variety of star formation histories (SFHs) of $z\gtrsim6$ galaxies
provides important insights into early star formation, but has been difficult
to systematically quantify. Some observations suggest that many $z\sim6-9$
galaxies are dominated by $\gtrsim200$ Myr stellar populations, implying
significant star formation at $z\gtrsim9$, while others find that most
reionization era galaxies are $\lesssim10$ Myr, consistent with little
$z\gtrsim9$ star formation. Here, we quantify the distribution of ages of
UV-bright ($-22.5\lesssim M_{UV}\lesssim-21$) galaxies colour-selected to lie
at $z\simeq6.6-6.9$, an ideal redshift range to systematically study the SFHs
of reionization era galaxies with ground-based observatories and Spitzer. We
infer galaxy properties with two SED modelling codes and compare results,
finding that stellar masses are largely insensitive to the model, but the
inferred ages can vary by an order of magnitude. We infer a distribution of
ages assuming a simple, parametric SFH model, finding a median age of
$\sim30-70$ Myr depending on SED model. We quantify the fractions of $\leq10$
Myr and $\geq250$ Myr galaxies, finding that these systems comprise $\sim15-30$
percent and $\sim20-25$ percent of the population, respectively. With a
flexible SFH model, the shapes of the SFHs are consistent with those implied by
the simple model (e.g. young galaxies have rapidly rising SFHs). However,
stellar masses can differ significantly, with those of young systems sometimes
being more than an order of magnitude larger with the flexible SFH. We quantify
the implications of these results for $z\gtrsim9$ stellar mass assembly and
discuss improvements expected from JWST.
