Learn English – Why use ‘step down’ instead of ‘resign’? Is there any difference

The words are similar, but not close enough to be synonyms. To plumb something means:

to try to understand or succeed in understanding something mysterious

This is a very literary usage. Further, if you plumb the depths of something you are saying that:

to be or to experience an extreme example of something unpleasant

I think you're mixing phrasal verbs: to track something down means to go look for something. To plumb something means that you are trying to understand it. But you shouldn't use plumb something down because that is just a mix of the two. So your second sentence should be:

I've plumbed the problem and found something blocking our way.

To plumb something is not likely to be as fully understood as "track". As the dictionary says, it is more of a literary usage. Stick to tracking something down--it will be more understood, which in a programming context is important.

Oishi-san, it works this way in the political calendar:

An August pick happens around the time of the political conventions, and is calculated to arouse the party base.

An October pick is calculated to be a strong campaigner right before the election, which takes place on the first Tuesday of the following month.

A January pick, however, is someone who would be a good and competent Vice President after the inauguration. One supposes the January pick would be chosen only if the president needed no help to win the election, and was perceived as a shoo-in — the way Nixon was in 1972, say.