
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
Let  $F_k(a,b)=(a+b)^k-a^k-b^k$  and let  $S={1,2,3,4,5,6,7,8,9,10}$ . For how many ordered pairs  $(a,b)$  with  $a,b\in S$  and  $a\leq b$  is  $\frac{F_5(a,b)}{F_3(a,b)}$  an integer?

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