OK, you don't seem to understand Turkey doesn't have the financial means, you can't just develop an absolutely high end engine and produce it on an assembly line just because you or I want it that way. The aircraft also has to be affordable, both in terms of the purchase price for the armed forces and the sales price for export.Neither the F110 nor the F119 is an absolutely high-end engine right now. Turkiye can develop and produce an engine better than F110.
If it's more expensive than an F-35/J-35, you've already lost the export market.
The more complex the engine is, the more expensive it is to manufacture and maintain.
Turkey does not have a yearly military budget of 800-900 billion dollars like the US, which develops weapon systems/engines without compromise if it has to.
Therefore, Turkey must first develop an engine that surpasses the upper performance class, in this case the F-110/EJ200.
The dimensions of the F-110 in TAI KAAN are the fixed benchmark, so you can't just change the engine at will, but there are limits set by the size of the F-110.
The F-110 engine has 3 main fans and 9 compressors as well as a low pressure turbine and a high pressure turbine.
One could reduce one of the main fans and one compressor and use 2 low- and high-pressure turbines instead.
The result is an engine with significantly fewer mechanically moving parts. The complexity of the engine is lower, as is the manufacturing and maintenance, and it is still slightly more powerful than an F-110/EJ200. You can still incorporate modern approaches like VCT & Co.
If you then have variable air intakes in the TAI KAAN, you can also trigger a RAM effect and get additional thrust, on average up to 10% more thrust is possible, e.g. for supercruise without afterburner or better thrust in the subsonic range.