
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 $f(k)$ be the number of natural numbers $x$ that satisfy the inequality $\log_{2}x + \log_{2}(5 \cdot 2^{k-1} - x) \geqslant 2k$, where $k \in \mathbb{N}^*$.

1. Find the functional expression for $f(k)$.
2. Let $S_{n} = f(1) + 2f(2) + \dots + nf(n)$. Find $S_{n}$.

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