
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 the positive value of  $ a$  such that the curve  $ C_1: x \equal{} \sqrt {2y^2 \plus{} \frac {25}{2}}$  tangent to the parabola  $ C_2: y \equal{} ax^2$ , then find the equation of the tangent line of  $ C_1$  at the point of tangency.

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