
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
In triangle \( \triangle ABC \), sides \( BC \) and \( AC \) have fixed lengths \( a \) and \( b \) respectively, while the length of side \( AB \) can vary. A square is constructed outward on side \( AB \). Let \( O \) be the center of the constructed square, and let \( M \) and \( N \) be the midpoints of \( BC \) and \( AC \) respectively. Find the maximum value of \( OM + ON \).

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