
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
Define a new sequence $G_n$ where $G_1 = 1, G_2 = 1$ and $G_n = G_{n-1} + G_{n-2} + 1$ for all $n \ge 3$. Given that the numbers $G_a, G_b, G_c$ form an increasing geometric sequence and $a + b + c = 2100$, find the value of $a$.

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