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how to obtain p values from an ANOVA result

It depends on what type of model "model" is.

In some cases you can do something like
anova(model, test='Chisq')

See the appropriate help to find out the options, but note that if the p-values are not provided by default, it could be because there is doubt about the accuracy of the approximation and you should use them with caution (or use simulation to verify the quality of the approximation). 

Hope this helps,