
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
Let  $a$  be a positive number. Assume that the parameterized curve  $C: \ x=t+e^{at},\ y=-t+e^{at}\ (-\infty <t< \infty)$  is touched to  $x$  axis.

(1) Find the value of  $a.$ 
(2) Find the area of the part which is surrounded by two straight lines  $y=0, y=x$  and the curve  $C.$ 

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