
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 a particular value of the angle  $\theta$  we can take the product of the two complex numbers  $(8+i)\sin\theta+(7+4i)\cos\theta$  and  $(1+8i)\sin\theta+(4+7i)\cos\theta$  to get a complex number in the form  $a+bi$  where  $a$  and  $b$  are real numbers.  Find the largest value for  $a+b$ .

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