
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 a quadrilateral  $ABCD$ , we have  $\angle DAB = 110^{\circ} , \angle ABC = 50^{\circ}$  and  $\angle BCD = 70^{\circ}$  . Let  $ M, N$  be the mid-points of  $AB$  and  $CD$  respectively. Suppose  $P$  is a point on the segment  $M N$  such that  $\frac{AM}{CN} = \frac{MP}{PN}$  and  $AP = CP$  . Find  $\angle AP C$ .

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