
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 one side of an angle with vertex at point $M$, measuring $30^\circ$, there are points $A$ and $B$ such that $MA = 1$ and $MB = 3$. A point $C$ moves along the other side of the angle. Find $MC$ given that:

a) The angle $BCA$ is the greatest;

b) The sum of the squares of the distances from $C$ to points $B$ and $A$ is the smallest;

c) The sum of the distances from $C$ to points $B$ and $A$ is the smallest;

d) The trajectory BSM is optimal when moving from point $B$ to point $M$ given that movement along the ray containing point $C$ is twice as fast as any other path outside this ray.

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