
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  $\triangle ABC$ ,  $AB = 30$ ,  $BC = 40$ , and  $CA = 50$ . Squares  $A_1A_2BC$ ,  $B_1B_2AC$ , and  $C_1C_2AB$  are erected outside  $\triangle ABC$ , and the pairwise intersections of lines  $A_1A_2$ ,  $B_1B_2$ , and  $C_1C_2$  are  $P$ ,  $Q$ , and  $R$ . Compute the length of the shortest altitude of  $\triangle PQR$ .

*Proposed by Lewis Chen*

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