
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
We form a right-angled grid with strips of width $2c$. The grid openings are congruent rectangles with sides $a$ and $b$ (where $a > b$). A ball with radius $r$ is thrown perpendicularly to the plane of the grid.

a) What is the probability that the ball will pass through the grid without touching it?

b) For a constant grid opening, what strip width should be chosen so that the above probability - for any ball - is always less than 1/2? Construct this strip width.

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