
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

$$
\frac{a}{b} = \frac{c}{d}
$$

and

$$
\frac{a_{1}}{b_{1}} = \frac{c_{1}}{d_{1}}
$$

what is the condition for

$$
\frac{a + a_{1}}{b + b_{1}} = \frac{c + c_{1}}{d + d_{1}}?
$$

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