
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 the curve $y = x^n + 1$ ($n \in \mathbb{N}^*$) have a tangent line at the point $(1,1)$, and let the x-coordinate of the point where this tangent line intersects the x-axis be $x_n$.  
Define $a_n = \lg x_n$, then the value of $a_1 + a_2 + \ldots + a_{99}$ is \_\_\_\_\_\_\_\_.

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