
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

Given a right triangle $\Delta A B_{0} C_{0}$ with a right angle at $B_{0}$. From point $B_{0}$, a perpendicular is drawn to $A C_{0}$; from the foot point $C_{1}$ of this perpendicular, another perpendicular is drawn to $A B_{0}$, reaching the foot point $B_{1}$, and this process is repeated indefinitely. The foot points of the perpendiculars drawn to $A C_{0}$ and $A B_{0}$ are $C_{1}, C_{2}, \ldots, C_{i}, \ldots$ and $B_{1}, B_{2}, \ldots, B_{i}, \ldots$, respectively. Calculate the following sums (from $i=0$ to $i=\infty$):

1. $\sum B_{i} C_{i}$
2. $\sum A B_{i}$
3. $\sum A C_{i}$
4. $\sum C_{i+1} B_{i}$

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