
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
If  $a$  and  $b$  are complex numbers such that  $a^2 + b^2 = 5$  and  $a^3 + b^3 = 7$ , then their sum,  $a + b$ , is real. The greatest possible value for the sum  $a + b$  is  $\tfrac{m+\sqrt{n}}{2}$  where  $m$  and  $n$  are integers. Find  $n.$ 

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