
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

For each non-right triangle \( T \), let \( T_1 \) be the triangle formed by the feet of the altitudes of \( T \); let \( T_2 \) be the triangle formed by the feet of the altitudes of \( T_1 \); similarly define the triangles \( T_3, T_4 \), and so on. What must triangle \( T \) be such that,

a) Triangle \( T_1 \) is acute-angled?

b) In the sequence \( T_1, T_2, T_3, \ldots \), there appears a right triangle \( T_n \) (hence, triangle \( T_{n+1} \) is not defined)?

c) Triangle \( T_3 \) is similar to triangle \( T \)?

d) For each natural number \( n \), determine how many non-similar triangles \( T \) exist for which triangle \( T_n \) is similar to triangle \( T \)?

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