
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
12. What is the sum of all possible  $\left(\begin{array}{l}i  j\end{array}\right)$  subject to the restrictions that  $i \geq 10, j \geq 0$ , and  $i+j \leq 20$  ? Count different  $i, j$  that yield the same value separately - for example, count both  $\left(\begin{array}{c}10  1\end{array}\right)$  and  $\left(\begin{array}{c}10  9\end{array}\right)$ .

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