
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 $f(k)$ denote the number of natural numbers in the interval $[2^{k-1}, 2^k]$ ($k \in \mathbb{N}^*$). Let $S_n = f(1) + f(2) + \ldots + f(n)$.  
(1) Find the expression for $S_n$; (2) Let $P_n = n^2 + n - 1$ ($n \in \mathbb{N}^*$), compare the magnitude of $S_n$ and $P_n$, and explain the reason.

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