
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
Consider the following  $50$ -term sums: $S=\frac{1}{1\cdot 2}+\frac{1}{3\cdot 4}+...+\frac{1}{99\cdot 100}$ , $T=\frac{1}{51\cdot 100}+\frac{1}{52\cdot 99}+...+\frac{1}{99\cdot 52}+\frac{1}{100\cdot 51}$ .

Express  $\frac{S}{T}$  as an irreducible fraction.


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