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Galaxy clusters are the products of structure formation through myriadphysical processes that affect their growth and evolution throughout cosmichistory. As a result, the matter distribution within galaxy clusters, or theirshape, is influenced by cosmology and astrophysical processes, in particularthe accretion of new material due to gravity. We introduce an analysis methodto investigate the 3D triaxial shapes of galaxy clusters from the ClusterHEritage project with XMM-Newton -- Mass Assembly and Thermodynamics at theEndpoint of structure formation (CHEX-MATE). In this work, the first paper of aCHEX-MATE triaxial analysis series, we focus on utilizing X-ray data from XMMand Sunyaev-Zel'dovich (SZ) effect maps from Planck and ACT to obtain a threedimensional triaxial description of the intracluster medium (ICM) gas. Wepresent the forward modeling formalism of our technique, which projects atriaxial ellipsoidal model for the gas density and pressure to compare directlywith the observed two dimensional distributions in X-rays and the SZ effect. AMarkov chain Monte Carlo is used to estimate the posterior distributions of themodel parameters. Using mock X-ray and SZ observations of a smooth model, wedemonstrate that the method can reliably recover the true parameter values. Inaddition, we apply the analysis to reconstruct the gas shape from the observeddata of one CHEX-MATE galaxy cluster, Abell 1689, to illustrate the technique.The inferred parameters are in agreement with previous analyses for thatcluster, and our results indicate that the geometrical properties, includingthe axial ratios of the ICM distribution, are constrained to within a fewpercent. With much better precision than previous studies, we thus furtherestablish that Abell 1689 is significantly elongated along the line of sight,resulting in its exceptional gravitational lensing properties.

CHEX-MATE: CLUster Multi-Probes in Three Dimensions (CLUMP-3D), I. Gas Analysis Method using X-ray and Sunyaev-Zel'dovich Effect Data