
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
In cyclic quadrilateral  $ABXC$ ,  $\measuredangle XAB = \measuredangle XAC$ . Denote by  $I$  the incenter of  $\triangle ABC$  and by  $D$  the projection of  $I$  on  $\overline{BC}$ . If  $AI = 25$ ,  $ID = 7$ , and  $BC = 14$ , then  $XI$  can be expressed as  $\frac{a}{b}$  for relatively prime positive integers  $a, b$ . Compute  $100a + b$ .

*Proposed by Aaron Lin*

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