
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]


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On side \(AB\) of triangle \(ABC\), a square with center \(O\) is constructed externally. Points \(M\) and \(N\) are the midpoints of sides \(AC\) and \(BC\) respectively, and the lengths of these sides are \(a\) and \(b\) respectively. Find the maximum value of the sum \(OM + ON\) as the angle \(ACB\) varies.

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