
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
Suppose that  $a$  and  $b$  are real numbers such that  $\sin(a)+\sin(b)=1$  and  $\cos(a)+\cos(b)=\frac{3}{2}$ . If the value of  $\cos(a-b)$  can be written as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ , determine  $100m+n$ .

*Proposed by Michael Ren*

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