
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
Given a right-angled triangle \(ABC\) with legs \(AC = a\) and \(CB = b\), find:

a) the side length of the square (with one vertex at \(C\)) having the largest area that lies entirely within the triangle \(ABC\).

b) the dimensions of the rectangle (with one vertex at \(C\)) having the largest area that lies entirely within the triangle \(ABC\).

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