
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
Alice has an isosceles triangle  $M_0N_0P$ , where  $M_0P=N_0P$  and  $\angle M_0PN_0=\alpha^{\circ}$ . (The angle is measured in degrees.) Given a triangle  $M_iN_jP$  for nonnegative integers  $i$  and  $j$ , Alice may perform one of two *elongations*:

a) an  $M$ -*elongation*, where she extends ray  $\overrightarrow{PM_i}$  to a point  $M_{i+1}$  where  $M_iM_{i+1}=M_iN_j$  and removes the point  $M_i$ .
b)  an  $N$ -*elongation*, where she extends ray  $\overrightarrow{PN_j}$  to a point  $N_{j+1}$  where  $N_jN_{j+1}=M_iN_j$  and removes the point  $N_j$ .

After a series of  $5$  elongations,  $k$  of which were  $M$ -elongations, Alice finds that triangle  $M_kN_{5-k}P$  is an isosceles triangle. Given that  $10\alpha$  is an integer, compute  $10\alpha$ .

*Proposed by Yannick Yao*

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