
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  $ABC$  be a triangle with  $BC = 5$ ,  $CA = 3$ , and  $AB = 4$ . Variable points  $P, Q$  are on segments  $AB$ ,  $AC$ , respectively such that the area of  $APQ$  is half of the area of  $ABC$ . Let  $x$  and  $y$  be the lengths of perpendiculars drawn from the midpoint of  $PQ$  to sides  $AB$  and  $AC$ , respectively. Find the range of values of  $2y + 3x$ .

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