
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
In triangle \(ABC\), heights \(AA_1\) and \(BB_1\) are drawn. Find \(AC\) if:
a) \(AA_1 = 4\), \(BB_1 = 5\), \(BC = 6\)

b) \(A_1C = 8\), \(B_1C = 5\), \(BB_1 = 12\)

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