
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
On a plane, a line  $\ell$  and two circles  $c_1$  and  $c_2$  of different radii are given such that   $\ell$   touches both circles at point  $P$ . Point  $M \ne P$  on  $\ell$  is chosen so that the angle  $Q_1MQ_2$  is as large as possible where  $Q_1$  and  $Q_2$  are the tangency points of the tangent lines drawn from  $M$  to  $c_i$  and  $c_2$ , respectively, differing from  $\ell$  . Find  $\angle PMQ_1 + \angle PMQ_2$ ·

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