
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 the diagram, \(\triangle PQR\) is an isosceles triangle with \(PQ = PR\). Semi-circles with diameters \(PQ\), \(QR\), and \(PR\) are drawn. The sum of the areas of these three semicircles is equal to 5 times the area of the semi-circle with diameter \(QR\). Find the value of \(\cos (\angle PQR)\).

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