
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
From a point \( A \) on a horizontal plane, an object is thrown with a speed of \( c \, \text{m/s} \) at an angle \( \alpha \). Simultaneously, from a point \( B \) located \( a \, \text{m} \) horizontally away from point \( A \), another object is thrown vertically upwards with the same speed. After how many seconds will the distance between the two objects be the smallest? What is this minimum distance if \( \alpha = 30^\circ \), \( c = 50 \, \text{m/s} \), and \( a = 50 \sqrt{3} \, \text{m} \)?

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