
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 triangle  $ABC$ , let  $J$  be the center of the excircle tangent to side  $BC$  at  $A_{1}$  and to the extensions of the sides  $AC$  and  $AB$  at  $B_{1}$  and  $C_{1}$  respectively. Suppose that the lines  $A_{1}B_{1}$  and  $AB$  are perpendicular and intersect at  $D$ . Let  $E$  be the foot of the perpendicular from  $C_{1}$  to line  $DJ$ . Determine the angles  $\angle{BEA_{1}}$  and  $\angle{AEB_{1}}$ .

*Proposed by Dimitris Kontogiannis, Greece*

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