
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 us draw the height BH.  
$\angle \mathrm{BCH} = \angle \mathrm{CBH} = 45^{\circ}$, $\angle \mathrm{ABH} = 60^{\circ}$. Let $\mathrm{BH} = \mathrm{HC} = x$, then $\mathrm{BC} = x \sqrt{2}$, $\mathrm{AB} = 2x$.

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

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