The most intensively star-forming galaxies are extremely luminous at far-infrared (FIR) wavelengths, highly obscured at optical and ultraviolet wavelengths, and lie at $zge 1-3$. We present a programme of ${it Herschel}$ FIR spectroscopic observations with the SPIRE FTS and photometric observations with PACS, both on board ${it Herschel}$, towards a sample of 45 gravitationally lensed, dusty starbursts across $zsim 1-3.6$. In total, we detected 27 individual lines down to 3-$sigma$, including nine $[rm C{small II}]$ 158-$mu$m lines with confirmed spectroscopic redshifts, five possible $[rm C{small II}]$ lines consistent with their far-infrared photometric redshifts, and in some individual sources a few $[rm O{small III}]$ 88-$mu$m, $[rm O{small III}]$ 52-$mu$m, $[rm O{small I}]$ 145-$mu$m, $[rm O{small I}]$ 63-$mu$m, $[rm N{small II}]$ 122-$mu$m, and OH 119-$mu$m (in absorption) lines. To derive the typical physical properties of the gas in the sample, we stack all spectra weighted by their intrinsic luminosity and by their 500-$mu$m flux densities, with the spectra scaled to a common redshift. In the stacked spectra, we detect emission lines of $[rm C{small II}]$ 158-$mu$m, $[rm N{small II}]$ 122-$mu$m, $[rm O{small III}]$ 88-$mu$m, $[rm O{small III}]$ 52-$mu$m, $[rm O{small I}]$ 63-$mu$m, and the absorption doublet of OH at 119-$mu$m, at high fidelity. We find that the average electron densities traced by the $[rm N{small II}]$ and $[rm O{small III}]$ lines are higher than the average values in local star-forming galaxies and ULIRGs, using the same tracers. From the $[rm N{small II}]/[rm C{small II}]$ and $[rm O{small I}]/[rm C{small II}]$ ratios, we find that the $[rm C{small II}]$ emission is likely dominated by the photo-dominated regions (PDR), instead of by ionised gas or large-scale shocks.
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