
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  $\ell$  be a line in the plane and let  $90^\circ<\theta<180^\circ$ . Consider any distinct points  $P,Q,R$  that satisfy the following: 

(i)  $P$  lies on  $\ell$  and  $PQ$  is perpendicular to  $\ell$ (ii)  $R$  lies on the same side of  $\ell$  as  $Q$ , and  $R$  doesn’t lie on  $\ell$ (iii) for any points  $A,B$  on  $\ell$ , if  $\angle ARB=\theta$  then  $\angle AQB \geq \theta$ . 

Find the minimum value of  $\angle PQR$ .

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