
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
The planes $S$ and $T$ are perpendicular to each other, sharing a common line $m$. Each plane contains a point, $C$ in $S$ and $D$ in $T$. The projections of these points onto the line $m$ are $C^{\prime}$ and $D^{\prime}$, respectively. Given that $C C^{\prime}=c$, $D D^{\prime}=d$, and $C^{\prime} D^{\prime}=e$, construct two spheres of equal radius that touch each other, with one sphere touching $S$ at $C$ and the other touching $T$ at $D$.

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