## 2 Materials and Methods  
In order to prepare the electrolyte with the composition of La\({}_{0.8}\)Sr\({}_{0.2}\)Ga\({}_{0.8}\)Mg\({}_{0.2}\)O\({}_{3-\delta}\) (LSGM), a conventional solid-state method was applied using La\({}_{2}\)O\({}_{3}\) (99.9 wt.%), SrCO\({}_{3}\) (99.9 wt.%), Ga\({}_{2}\)O\({}_{3}\) (99.9 wt.%) and MgO (99.5 wt.%). The starting compounds were mixed in isopropanol and ground in a PM 100 planetary ball mill (Retsch\({}^{\copyright\)}\), Haan, Germany), and then dried and annealed at 1000 \({}^{\circ}\)C for 6 h in air, followed by reginding in the ball mill. The resulting powder was pressed and sintered for 6 h to obtain dense ceramic samples and the sintering temperatures were 1350, 1400, and 1450 \({}^{\circ}\)C. For the samples with a sintering aid, 0.5 wt.% Fe\({}_{2}\)O\({}_{3}\) (99.9%) was added to the powder before pressing and mixed in the planetary mill followed by sintering. Similarly, the solid-state method and the same starting compounds were used in the synthesis of La\({}_{0.6}\)Sr\({}_{0.4}\)FeO\({}_{3-\delta}\) (LSF) electrode powder. The first synthesis step was performed at 900 \({}^{\circ}\)C for 3 h, and the final sintering step was performed at 1150 \({}^{\circ}\)C for 2 h in air. The electrode powder was then ground to an average particle size of about 5 \(\upmu\)m with the addition of sintering aids of 3 wt.% Fe\({}_{2}\)O\({}_{3}\) or 3 wt.% CuO.  
The samples for high-temperature measurements were prepared in the button type with symmetrical electrodes and the supporting LSGM electrolyte. In order to measure the performance of the electrolyte, the electrodes were made of platinum and sintered at 1050 \({}^{\circ}\)C for 2 h. To evaluate the performance of the electrodes, LSF electrodes were fabricated on the base of electrolyte buttons. The electrode ink, which was prepared by mixing an organic binder with the LSF powder, was applied to both sides of LSGM using the print-screen method and sintered at temperatures of 1050, 1100, and 1150 \({}^{\circ}\)C.  
The X-ray powder diffraction analysis was carried out using a D/MAX-2200 diffractometer (Rigaku Corporation, Takatsuki, Japan) with CuK\({}_{\alpha}\)-radiation (\(\lambda\)(K\({}_{\alpha}\)) = 1.5406 A, 40 kV, 30 mA) at room temperature in ambient air. The microstructure of the electrolyte and electrolyte/electrode boundary was analyzed by scanning electron microscopy (SEM) using MIRA 3 (Tescan, Brno, Czech Republic) and KYKY-EM8200 (KYKY Technology Co., Ltd., Beijing, China) electron microscopes.  
An investigation of the high-temperature performance of the LSGM electrolyte was carried out using impedance spectroscopy with the P-40X instrument (Elims, Chernogolovka, Russia) over a temperature range of 200-550 \({}^{\circ}\)C in atmospheres with various oxygen partial pressures (pO\({}_{2}\) = 0.21 and 10\({}^{-22}\) atm). A high-temperature Pt/YSZ/Pt pump, located in a discrete furnace and operating at 800 \({}^{\circ}\)C, was used to remove oxygen from the air. The contributions of the bulk and grain boundary resistance to the total resistance of the LSGM electrolyte were determined using ZView software, version 2.80 (Scribner Associates Inc., Southern Pines, NC, USA).  
The electrochemical performance of LSF electrodes was investigated through impedance spectroscopy, using a Solartron FRA-1260 and EI-1287 (Ametek, Hampshire, UK) in both air and wet (3 vol.% H\({}_{2}\)O) hydrogen atmospheres. For the distribution of relaxation time (DRT) analysis of the impedance spectra, the original software based on [PERSON]'s regularization was utilized.