
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 a parallelogram  $ABCD$ , let  $\mathcal{P}$  be a plane such that the distance from vertex  $A$  to  $\mathcal{P}$  is  $49$ , the distance from vertex  $B$  to  $\mathcal{P}$  is  $25$ , and the distance from vertex  $C$  to  $\mathcal{P}$  is  $36$ . Find the sum of all possible distances from vertex  $D$  to  $\mathcal{P}$ .

*Proposed by **HrishiP***

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