
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
Given the parallelogram  $ABCD$ . The circle  $S_1$  passes through the vertex  $C$  and touches the sides  $BA$  and  $AD$  at points  $P_1$  and  $Q_1$ , respectively. The circle  $S_2$  passes through the vertex  $B$  and touches the side  $DC$  at points  $P_2$  and  $Q_2$ , respectively. Let  $d_1$  and  $d_2$  be the distances from  $C$  and  $B$  to the lines  $P_1Q_1$  and  $P_2Q_2$ , respectively.
Find all possible values of the ratio  $d_1:d_2$ .

*(I. Voronovich)* 

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