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Active galactic nuclei (AGN) are variable on different timescales which scale linearly with the mass of the central black hole. Optical variability is considered among the most reliable selection criteria for unobscured AGN. The light curve is a time series of flux measurements, which can be derived from archival images. In this work, we consider a sample of more than 300 relatively weak AGN powered by "light-weight" supermassive black holes (SMBH) with masses up to 1e6 solar masses selected from archival X-ray observations and with the mass estimates from optical spectroscopy (H-alpha broad line with and flux). We use Zwicky Transient Facility (ZTF) data to detect optical variability at a few percent scale. ZTF Data Release aperture light curves cannot be used for our purpose because of systematic errors reaching 15-20% making detection of weak variability on top of bright host galaxies unfeasible. We describe how to build reliable light curves using the ZTF Forced-Photometry service, which uses difference images for flux estimates. Such light curves are much better already "off-the-shelf", because they are almost free of systematic errors introduced by the host galaxies and variable seeing conditions, however we need extra post-processing / filtering steps to further improve them and get rid of the remaining systematics. We set the threshold signal-to-noise ratio (SNR) to 3, points with a lower SNR were excluded. The distance to the nearest reference was set to less than 1 arcsec, pixel quality indicators helped to clear outliers related to uncorrected cosmic ray hits and hot pixels. The zero point correction was made for each observation, based on the CCD quadrant, and the average airmass of observations. A color correction was made using the zero color from the Pan-STARRS1 photometric system. For each filter, the ZTF field, CCD quadrant the light curves were studied. Then we defined the level of significance according to the chi-square criterion at which the light curve is not consistent with white noise on top of a constant. Many of the studied objects showed statistically significant variability on the timescales from days to months. Also, a strong flare with an amplitude of 0.25 mag was observed in the object J112637.74+513423.0 for about 3 months, which can be either a heavily dust obscured supernova explosion or a star rupture by tidal forces in the SMBH potential (tidal disruption event). The bottleneck limiting the scalability of our analysis and making it yet impossible to study tens of thousands of AGN is that the ZTF Forced-Photometry Service can process up to 100 objects at a time and it takes hours to weeks to process the data. Therefore, we first process the most astrophysically interesting sources then gradually expanding the sample.