
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
For postive constant  $ r$ , denote  $ M$  be the all sets of complex  $ z$  that satisfy  $ |z \minus{} 4 \minus{} 3i| \equal{} r$ .

(1) Find the value of  $ r$  such that there is only one real number which is belong to  $ M$ .

(2) If  $ r$  takes the value found in (1), find the maximum value  $ k$  of  $ |z|$  for complex number  $ z$  which is belong to  $ M$ .

Present the answer in LaTex format: \boxed{Your answer}
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