ーながら ＆ できる

In Japanese we for 'can' with a verb we use the following grammar (replace verb with one of your choice in dictionary form):

…行くことができる。

But we can use a shorter form - though this change depends on what type of verb we are using, i.e. group I, II or III. In this example, with a group I verb we use ける replacing the く in 行く. Then, the sentence above becomes a single word with the same meaning:

…行ける。

For describing 'while' events we use ーながら after the ます form of a verb. When translating I sometimes get confused which is the 'while' verb as I studied that the second verb is the most important while some sentence translations I've seen seem to emphasize the first verb:

私はテレビを見ながら夕飯を食べる。

So this is all well and good, but today I tried the following sentence:

1. 山中湖で富士山が見えながらテニスをした。

My intent was to try and combine:

2. 富士山が見える & 3. 富士山を見ながら…

So, the end result: you can't make sentences like 1! Two sentences are needed for my intended meaning. So there you are, don't try and combine these two grammar structures.