We present the AGILE-GRID monitoring of Cygnus X-3, during the period between November 2007 and July 2009. We report here the whole AGILE-GRID monitoring of Cygnus X-3 in the AGILE pointing mode data-taking, to confirm that the gamma-ray activity coincides with the same repetitive pattern of multiwavelength emission and to analyze in depth the overall gamma-ray spectrum by assuming both leptonic and hadronic scenarios. Seven intense gamma-ray events were detected in this period, with a typical event lasting one or two days. These durations are longer than the likely cooling times of the gamma-ray emitting particles, implying we see continuous acceleration rather than the result of an impulsive event such as the ejection of a single plasmoid which then cools as it propagates outwards. Cross-correlating the AGILE-GRID light curve with X-ray and radio monitoring data, we find that the main events of gamma-ray activity have been detected while the system was in soft spectral X-ray states (RXTE/ASM count rate > 3 counts/s), that coincide with local and often sharp minima of the hard X-ray flux (Swift/BAT count rate < 0.02 counts/cm^2/s), a few days before intense radio outbursts. [...] These gamma-ray events may thus reflect a sharp transition in the structure of the accretion disk and its corona, which leads to a rebirth of the microquasar jet and subsequent enhanced radio activity. [...] Finally, we examine leptonic and hadronic emission models for the gamma-ray events and find that both scenarios are valid. In the leptonic model - based on inverse Compton scatterings of mildly relativistic electrons on soft photons from the Wolf-Rayet companion star and from the accretion disk - the emitting particles may also contribute to the overall hard X-ray spectrum, possibly explaining the hard non-thermal power-law tail sometimes seen during special soft X-ray states in Cygnus X-3.
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