
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
Let \( n \) be a positive integer. Find the following three sums:

\[ 
\begin{array}{l}
S_{n} = \mathrm{C}_{n}^{0} + \mathrm{C}_{n}^{3} + \cdots + \mathrm{C}_{n}^{3k} + \cdots, \\
T_{n} = \mathrm{C}_{n}^{1} + \mathrm{C}_{n}^{4} + \cdots + \mathrm{C}_{n}^{3k+1} + \cdots, \\
H_{n} = \mathrm{C}_{n}^{2} + \mathrm{C}_{n}^{5} + \cdots + \mathrm{C}_{n}^{3k+2} + \cdots.
\end{array}
\]

It is given that when \( m > n \), \( \mathrm{C}_{n}^{m} = 0 \).

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