There are two main reasons why this statement is incorrect: First, most of the Red Army (but the Shaanxi-Gansu detachment) led by Chairman Mao crossed Minshan in September 1935, and Chairman Mao wrote this poem in June 1935. At this time, it is still in Songpan Grassland in Shan Zhinan, the Red Fourth Army south of Minshan Mountain. It doesn't go north through Minshan Mountain, but goes south. Secondly, the Red Second Army was founded on July 5, 1936, 9 months later than Chairman Mao wrote this poem. Obviously, it is impossible for Chairman Mao to write something that is not in the poem.
The correct interpretation of the term "three services" should be: "generally refers to the Red Army". Because the poem "Long March" is an ancient poem written by Chairman Mao, in ancient Chinese, "three armies" has four meanings, which are explained in detail and exemplified in Ci Yuan, and the second of the four meanings in Ci Yuan is "the general name of the army".
"Do" means all, all.