
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
Determine all infinite strings of letters with the following properties:
(a) Each letter is either \(T\) or \(S\),
(b) If position \(i\) and \(j\) both have the letter \(T\), then position \(i + j\) has the letter \(S\),
(c) There are infinitely many integers \(k\) such that position \(2k-1\) has the \(k\)th \(T\).

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