
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
Given two positive integers  $m$  and  $n$ , find the largest  $k$  in terms of  $m$  and  $n$  such that the following condition holds: 

Any tree graph  $G$  with  $k$  vertices has two (possibly equal) vertices  $u$  and  $v$  such that for any other vertex  $w$  in  $G$ , either there is a path of length at most  $m$  from  $u$  to  $w$ , or there is a path of length at most  $n$  from  $v$  to  $w$ .

*Proposed by Ivan Chan*

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