
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
Draw the height BH. \(\angle BCH = \angle CBH = 45^\circ\), \(\angle ABH = 60^\circ\). Let BH = HC = \(x\), then BC = \(x \sqrt{2}\), AB = \(2 x\).

\(\frac{AB}{BC} = \frac{BC}{MB} = \sqrt{2}\), \(\angle ABC\) is common. Therefore, \(\triangle MBC \sim \triangle CBA\) by the second criterion of similarity. From this similarity, it follows that \(\angle BMC = 45^\circ\). Then \(\angle AMC = 135^\circ\).

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