
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  $ABCD$  be a convex quadrilateral with  $\angle DAB =\angle B DC = 90^o$ . Let the incircles of triangles  $ABD$  and  $BCD$  touch  $BD$  at  $P$  and  $Q$ , respectively, with  $P$  lying in between  $B$  and  $Q$ . If  $AD = 999$  and  $PQ = 200$  then what is the sum of the radii of the incircles of triangles  $ABD$  and  $BDC$  ?

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