Section 10A. If the board with which a petition for the abatement of an assessment has been duly filed in accordance with the provisions of section five fails to act upon said petition within four months of the date of the filing of such petition, the petition shall be deemed to be denied, and the petitioner shall have the right within sixty days after the expiration of said four months to appeal as if the board had in fact denied the said petition; provided, that if the assessment has been paid, no appeal shall be taken after the expiration of ten months from the time of payment.