
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
Find all values of  $ \alpha$  for which the curves  $ y\equal{}\alpha x^2\plus{}\alpha x\plus{}\frac1{24}$  and  $ x\equal{}\alpha y^2\plus{}\alpha y\plus{}\frac1{24}$  are tangent to each other.

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