  MS 0735.6+7421 is a galaxy cluster which hosts a central radio galaxy with a
very steep spectrum, produced by one of the most powerful known jetted active
galactic nuclei (AGN). The radio plasma, ejected at nearly light speed from the
central AGN, have displaced the intra-cluster medium, leaving two pairs of
cavities observable in the X-ray, associated to two different outbursts, and
have distributed energy to the surrounding medium. In this work we have
performed for the first time a detailed, high-resolution spectral study of the
source at radio frequencies and investigated its duty cycle to be compared with
previous X-ray estimates. We have used new observations at 144 MHz produced
with the LOw Frequency ARray (LOFAR) together with archival data at higher
frequencies. At LOFAR frequency, the source presents two large outer radio
lobes, wider than at higher frequencies, and a smaller Intermediate lobe
located south-west of the core. A new inspection of X-ray data, allowed us to
identify an intermediate cavity, associated with that lobe, indicating the
presence of a further phase of jet activity. The radio lobes have a steep
spectrum even at LOFAR frequencies, reaching $\alpha_{144}^{610}=2.9$ in the
outer lobes and $\alpha_{144}^{610}=2.1$ in the Intermediate lobe. Fitting the
lobe spectra using a single injection model of particle ageing, we derived a
total age of the source between 170 and 106 Myr, in agreement with the buoyancy
and sound crossing time-scales derived from X-ray data. We then reconstructed
the duty cycle of the source. There were three phases of jet activity, with the
AGN being active for most of the time with only brief quiescent phases,
ensuring the repeated heating of the central gas. Finally, energetic estimates
revealed that a source of additional pressure support must be present to
sustain the bubbles against the pressure of the external medium.
