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Former Yeovil and Bristol City goalkeeper Weale spent the last three years at Leicester before being released by the Championship club at the end of last season.
The 30-year-old, along with Anyon, will work under goalkeeping coach Gavin Ward, himself a former Leicester goalkeeper, at Greenhous Meadow.
Anyon was player of the year at Conference club Lincoln last season.
The 25-year-old was with the Imps for two years, but made over 100 appearances for Port Vale prior to his move to Sincil Bank.
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