
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  $ABC$  be a triangle with  $BC = a$ ,  $CA = b$ , and  $AB = c$ . The  $A$ -excircle is tangent to  $\overline{BC}$  at  $A_1$ ; points  $B_1$  and  $C_1$  are similarly defined.

Determine the number of ways to select positive integers  $a$ ,  $b$ ,  $c$  such that


- the numbers  $-a+b+c$ ,  $a-b+c$ , and  $a+b-c$  are even integers at most 100, and
- the circle through the midpoints of  $\overline{AA_1}$ ,  $\overline{BB_1}$ , and  $\overline{CC_1}$  is tangent to the incircle of  $\triangle ABC$ .


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