Need help wiring Nest Gen3 , highly appreciated thank you !

I live in a high rise building constructed in the 1960s. The building has a pipe system, where there is a boiler and a chiller in the basement, and the building turns over from heat to cool based on the season. I have no control over heat or cold, only the fan in my condo which blows air over the heated/chilled pipe. All I have to control the unit is a three position switch for high speed / off / low speed. I'm trying to connect an Ecobee3 Lite to automate the process of turning the blower on and off based on the temperature in the room. I fully expect to lose the ability to choose high and low speed, and that's fine. I'm perfectly happy running the blower at high speed only. I'm doing a full renovation, and I had my contractor run a 5-wire thermostat cable from the HVAC, under the floor, and through the wall in advance of getting to this point. Unfortunately, neither my GC, nor his electrical subcontractor know how to make this system work. I've tried to use this post as a guide, as my building operates similarly, but to no avail. [url]https://www.doityourself.com/forum/thermostatic-controls/609404-getting-ecobee3-lite-work-old-fan-coil-system.html[/url] The fan controls in the post are a bit different than mine, and I can't quite figure out how to get from the other user's setup to my own. I've also reached out to my building engineer, who suggested a Honeywell fan center [url]https://www.supplyhouse.com/Honeywell-R8239A1052-40-VA-Fan-Center-w-SPDT-Switching-Action-includes-R8222B[/url], but when he tried to install it, the blower turned on right away, without even putting the Ecobee head unit in place. He then tried a different configuration, and the Honeywell just buzzed like it was going to explode when we turned the power on. I think this might be an appropriate solution, route, but I would need to try the installation myself given the previous experience with the building engineer. The manual is a bit overwhelming though: [url]https://s3.amazonaws.com/s3.supplyhouse.com/manuals/1262201469514/22892_PROD_FILE.pdf[/url] Alternatively, I purchased a 24V Transformer: [url=https://www.amazon.com/gp/product/B075PN6NCV/ref=ppx_yo_dt_b_asin_title_o00_s01?ie=UTF8&psc=1]https://www.amazon.com/gp/product/B075PN6NCV/ref=ppx_yo_dt_b_asin_title_o00_s01?ie=UTF8&psc=1[/url] to power the Ecobee, but it's not recognizing the switch. So basically, I have a lot of options, but I'm a little short of knowledge. I feel like I'm close though... [b]Here's what I'm working with: [/b] In the picture showing the wiring into the switch, there are two batches of wiring. The near batch has only two wires - red and white. Both are heavy gauge solid copper. The far batch has three wires - red, white, and blue, all braided copper. There are a lot more wires connecting to the motor, but I can't tell what anything else is connected to. I'm not entirely sure those other wires matter, so I'll leave those out of the equation for now. So all that said, here's an explanation of connections I can see: Near red (solid copper) to center position of the switch Near white (solid copper) to far white (braided) High speed terminal to braided red in far batch Low speed terminal to braided blue in far batch Any thoughts on how to wire all this up through the Ecobee would be much appreciated. Thank you in advance! [i]Note the "near" wiring on the left and the "far" on the right. The "near" batch is solid copper, the "far" is braided.[/i]